Scott McTominay's goal was Manchester United's 2,000th in the Premier League
Manchester United won a Premier League away game for the first time since February and would have beaten Norwich by more if not for two Tim Krul penalty saves.
United's two first-half goals came moments after outstanding Krul stops.
Scott McTominay slammed home the opener seconds after a stunning reaction save from the Canaries keeper to deny Anthony Martial from close range.
The video assistant referee (VAR) then controversially awarded United a penalty for a Ben Godfrey foul on Daniel James - with Krul stopping Marcus Rashford's kick.
But Rashford scored with United's next attack as he was left in acres of space by the Norwich defence.
United - who move up to seventh in the Premier League - were awarded a second penalty by VAR for a Todd Cantwell handball. This time Martial took it and went the other way, and again Krul guessed correctly.
The Canaries, who remain in the relegation zone, had a couple of chances but United wrapped the game up when Martial clipped the ball over Krul after a good passing move with Rashford.
Norwich scored a late consolation through substitute Onel Hernandez, who took the ball off McTominay before running at United's defence and shooting past David de Gea to becoming the first Cuban-born player to score a Premier League goal.
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Sunday's Premier League games as they happened
More disagreement on VAR use in the Premier League on Saturday
Krul and VAR catch the eye
In the first nine rounds of Premier League games this season no penalties were awarded by VAR with a seemingly high bar set for the "clear and obvious error" needed to overturn a referee's decision.
There have been four this weekend, including two at Carrow Road.
The first was highly controversial with Godfrey and James appearing to run into each other and referee Stuart Attwell changing his decision after consulting with the video team.
Krul appeared well off his line when he saved Martial's kick - but VAR in the Premier League does not check for goalkeepers' positions from penalties.
The second penalty was not contentious with the new handball laws with Cantwell's arms clearly high in the air as he blocked Fred's cross.
Martial became the third United player to fail from the spot in the league this season when Krul denied him.
He is the first goalkeeper to save two penalties in a Premier League game since fellow Dutchman Maarten Stekelenburg for Everton against Manchester City in October 2016.
Even United manager Solskjaer thought the first penalty should not have been given.
"I disagree with the first one, the second one is but both should be retaken, the goalkeeper is a yard off his line," he said.
"VAR is there to help but when it takes ages like the first penalty, it is not a clear and obvious error. When it took that long, it is a signal that it should not have been."
Tim Krul's stop from Anthony Martial's header is one of the saves of the season
Norwich manager Daniel Farke agreed, and thought United's James fouled Godfrey for the first penalty.
"It's quite obvious it was a big mistake," he told BBC Sport. "The referee was right and it was Daniel James falling on the defender. I don't blame VAR for this defeat, we were not major enough in the first half."
Krul was brilliant throughout the game, making eight saves - including a remarkable stop to keep out Martial's header from point-blank range.
He suffered a late knee injury, although played on with Norwich having used all their subs.
"We'll have to assess it tomorrow," he said after the game. "I was worried when it happened but I feel a bit better now."
United continue improvement
Martial is a very important player for us - Solskjaer
This is a third decent result in a row for Manchester United as they climb away from the lower reaches of the league.
A draw against leaders Liverpool last Sunday was followed by a first away win since Paris St-Germain in March, when they beat Partizan Belgrade 1-0 on Thursday.
A Norwich team short on confidence was an ideal opponent to end their worst away run in the league since 1989.
McTominay, Aaron Wan-Bissaka, Martial and Rashford were all excellent - despite the latter two's penalty misses.
With their first chance, Andreas Pereira curled a shot just wide and then Martial thought his header was in from Rashford's flick-on - before Krul's brilliant save.
They led when Scotland midfielder McTominay curled home after Cantwell failed to clear Pereira's cross - United's 2,000th goal in the Premier League era.
Only Rashford has scored more league goals for United than McTominay's four since Solskjaer was named permanent manager in March.
Rashford and Martial's penalty misses mean United have failed to score with four of their last six Premier League spot-kicks.
But that did not dishearten them with Rashford running onto James' defence-splitting pass to stroke home a shot past Krul.
Their third was excellent. Rashford played a one-two with Martial - on his first league start since August - and then backheeled the ball back into the Frenchman's path for his deft finish over Krul.
It is far too early to read too much into one good week for United, but things are looking a lot more encouraging than they were in the preceding five-game winless run.
"This had been a good week for us," said Solskjaer. "It started with a decent performance against Liverpool and now two wins away from home. It gives the players confidence and momentum.
"The third was a very good goal and a class finish. It is great for Anthony Martial to score a goal like that. We could and should have had a few more goals."
Long season ahead for Norwich
There was a minute's silence before the game for Norwich legend Duncan Forbes, who died on Wednesday
Norwich have now gone five games without a win in the league - and are two points below 17th-placed Newcastle.
They must be credited for trying to attack teams - right-back Max Aarons in particular was excellent going forward.
They will not struggle to create chances this season - Aarons set up good opportunities for Cantwell and Teemu Pukki, who both missed the target. Pukki has now gone five Premier League games without a goal, after six in his first five.
Hernandez's late strike after a McTominay error was their 11th Premier League goal of the season - more than sixth-placed Crystal Palace or Sheffield United in eighth.
But their defence was shambolic at times, especially when Rashford was completely free from James' pass over the top.
They have conceded 24 goals in 10 league games. Only Southampton - with 25 - have let in more, including the nine they conceded against Leicester. And it would have been worse if not for Krul.
Canaries boss Farke hopes things will be better when some of their injured centre-backs - Christoph Zimmermann, Timm Klose and Grant Hanley - return.
"In general it is difficult when three centre-halves are not available," he said.
"It's important to improve. We look forward to having our centre-backs back but we have to take this as a lesson.
"You need physicality and robustness, and we missed that in the first half."
Man of the match - Scott McTominay
Scott McTominay has become one of Manchester United's most important players. He covered more ground than any other player in the game (11.38km) and scored the opening goalRashford and Martial make history - match stats
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Norwich have lost more Premier League home games against Manchester United than they have against any other team, losing seven of nine (P9 W2 D0 L7).
Manchester United scored as many goals in this game as in their previous six Premier League away games combined.
Manchester United are the first team to score 2,000 Premier League goals.
Marcus Rashford's goal was his 50th for United, with 32 coming in the Premier League
Rashford and Anthony Martial are the first team-mates to score a goal and miss a penalty each in a Premier League game.
Norwich's Tim Krul became the eighth different goalkeeper in Premier League history to save two penalties in a game.
Manchester United have had three different players miss a penalty within a single Premier League season (Rashford, Martial and Paul Pogba) for only the second time, and for the first time since 2012-13 (Robin van Persie, Wayne Rooney and Javier Hernandez).
What's next?
Manchester United visit Chelsea in the Carabao Cup fourth round on Wednesday, 30 October (20:05 GMT), followed by a Premier League trip to Bournemouth on Saturday (12:30).
Norwich, already out of the cup, go to Brighton on Saturday (15:00).
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Line-ups
Match Stats
Live Text
Line-ups
Norwich
1Krul
2AaronsBooked at 90mins
24Amadou
4GodfreyBooked at 27mins
12LewisSubstituted forByramat 66'minutes
27Tettey
17Buendía
10LeitnerSubstituted forStiepermannat 45'minutes
23McLean
14CantwellSubstituted forHernándezat 45'minutes
22Pukki
Substitutes
3Byram
11Hernández
18Stiepermann
19Trybull
20Drmic
32Srbeny
33McGovern
Man Utd
1de Gea
29Wan-BissakaBooked at 45mins
2Lindelöf
5Maguire
18YoungBooked at 60mins
39McTominay
17Fred
21JamesSubstituted forLingardat 80'minutes
15PereiraSubstituted forGarnerat 82'minutes
10Rashford
9MartialSubstituted forGreenwoodat 75'minutes
Substitutes
8Mata
14Lingard
16Rojo
22Romero
26Greenwood
37Garner
53Williams
Referee:
Stuart Attwell
Attendance:
27,108
Match Stats
Home TeamNorwichAway TeamMan Utd
Possession
Home46%
Away54%
Shots
Home10
Away21
Shots on Target
Home3
Away11
Corners
Home1
Away10
Fouls
Home13
Away10
Live Text
Posted at
Match ends, Norwich City 1, Manchester United 3.
Full TimePosted at 90'+7'
Second Half ends, Norwich City 1, Manchester United 3.
Posted at 90'+4'
Attempt saved. Mason Greenwood (Manchester United) right footed shot from the centre of the box is saved in the bottom right corner. Assisted by Jesse Lingard.
Posted at 90'+3'
Attempt missed. Marcus Rashford (Manchester United) right footed shot from outside the box is high and wide to the left from a direct free kick.
BookingPosted at 90'+3'
Max Aarons (Norwich City) is shown the yellow card.
Posted at 90'+3'
James Garner (Manchester United) wins a free kick in the attacking half.
Posted at 90'+3'
Foul by Max Aarons (Norwich City).
Posted at 90'+1'
Attempt missed. Onel Hernández (Norwich City) right footed shot from outside the box is close, but misses to the left.
Goal!Posted at 88'
Goal! Norwich City 1, Manchester United 3. Onel Hernández (Norwich City) right footed shot from the centre of the box to the bottom right corner.
Posted at 85'
Foul by Fred (Manchester United).
Posted at 85'
Emiliano Buendía (Norwich City) wins a free kick in the defensive half.
Posted at 83'
Attempt missed. Jesse Lingard (Manchester United) right footed shot from outside the box is close, but misses the top right corner. Assisted by Harry Maguire.
SubstitutionPosted at 82'
Substitution, Manchester United. James Garner replaces Andreas Pereira.
SubstitutionPosted at 80'
Substitution, Manchester United. Jesse Lingard replaces Daniel James.
Posted at 78'
Scott McTominay (Manchester United) wins a free kick in the attacking half.
Posted at 78'
Foul by Onel Hernández (Norwich City).
Posted at 76'
Foul by Andreas Pereira (Manchester United).
Posted at 76'
Kenny McLean (Norwich City) wins a free kick in the defensive half.
SubstitutionPosted at 75'
Substitution, Manchester United. Mason Greenwood replaces Anthony Martial.
Goal!Posted at 73'
Goal! Norwich City 0, Manchester United 3. Anthony Martial (Manchester United) right footed shot from the left side of the six yard box to the centre of the goal. Assisted by Marcus Rashford.
Posted at 72'
Attempt saved. Onel Hernández (Norwich City) right footed shot from the left side of the box is saved in the bottom left corner.
Posted at 72'
Attempt blocked. Onel Hernández (Norwich City) right footed shot from the left side of the box is blocked. Assisted by Marco Stiepermann.
Posted at 68'
Foul by Scott McTominay (Manchester United).
Posted at 68'
Onel Hernández (Norwich City) wins a free kick in the defensive half.
Posted at 67'
Scott McTominay (Manchester United) wins a free kick in the defensive half.
Posted at 67'
Foul by Marco Stiepermann (Norwich City).
SubstitutionPosted at 66'
Substitution, Norwich City. Sam Byram replaces Jamal Lewis.
Posted at 66'
Attempt missed. Marcus Rashford (Manchester United) header from the centre of the box is too high. Assisted by Daniel James with a cross.
Posted at 63'
Corner, Manchester United. Conceded by Ben Godfrey.
Posted at 61'
Fred (Manchester United) wins a free kick in the defensive half.
Posted at 61'
Foul by Emiliano Buendía (Norwich City).
Posted at 60'
Attempt blocked. Onel Hernández (Norwich City) right footed shot from outside the box is blocked.
BookingPosted at 60'
Ashley Young (Manchester United) is shown the yellow card for a bad foul.
Posted at 59'
Foul by Ashley Young (Manchester United).
Posted at 59'
Max Aarons (Norwich City) wins a free kick on the right wing.
Posted at 58'
Foul by Fred (Manchester United).
Posted at 58'
Kenny McLean (Norwich City) wins a free kick in the defensive half.
Posted at 57'
Foul by Ashley Young (Manchester United).
Posted at 57'
Emiliano Buendía (Norwich City) wins a free kick on the right wing.
Posted at 54'
Attempt missed. Fred (Manchester United) left footed shot from outside the box is high and wide to the right following a corner.
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Source: bbc.com