Antonio Conte’s side were by far the better side and deserved more but the all-important breakthrough failed to materialise.
Phil Jagielka cleared off his own line twice in quick succession in the first half to deny Tiemoue Bakayoko and Willian while Ashley Williams headed against his own bar after the break.
Everton goalkeeper Jordan Pickford also made a couple of impressive saves to deny Pedro and Eden Hazard.
Conte made just one change to the side which beat Southampton a week ago, with Toni Rudiger coming into the defence at the expense of Gary Cahill.
Having missed the midweek Carabao Cup win over Bournemouth through suspension Marcos Alonsoreturned to the starting line-up, and with Alvaro Morata serving a one-match ban, it was a front three of Willian, Hazard and Pedro.
Having been an unused sub in midweek, 17-year-old Callum Hudson-Odoi was named on the bench in a Premier League game for the first time in his career.
There was no Wayne Rooney for Everton, who missed out due to illness, with Sam Allardyce naming an attacking trio of Aaron Lennon, Dominic Calvert-Lewin and Gylfi Sigurdsson
A nicely-worked move saw the Blues threaten inside the opening couple of minutes but, after the ball was played out to the right-hand side and Victor Moses delivered, Hazard, who had made a clever run to the near post, was unable to get enough of a connection on his volley.
It was all Chelsea during the opening stages, Everton were struggling to hold on and the home side had Jagielka to thank after 10 minutes as the skipper twice cleared off the line in quick succession, first to deny Bakayoko and then, from the follow-up, Willian, after Alonso’s volley had fizzed across the face of goal.
Hazard was drifting into space, picking up possession and beating his man at every opportunity, while N’Golo Kante and Bakayoko had a grip on the midfield area. As the 20-minute mark ticked by, however, it remained goalless and we had nothing to show for our dominance up until that point.
Cuco Martina became the second Everton player to enter the referee’s book, following Calvert-Lewin, after bringing Moses down just outside the box, and from the resulting free-kick Willian curled his effort over the bar
With 15 minutes remaining it looked as though the deadlock had been broken but as Williams diverted a Moses cross towards his own goal, the ball thumped against the underside of the bar before Everton cleared it away to safety. It was the closest we'd come to opening the scoring.
It was the Blues pressing for a goal as the clock ticked down and after a Hazard cross had caused panic and confusion in the penalty area, Fabregas's strike from distance took a deflection and rolled just wide.
Much like the first half, Everton had done very little of note in an attacking sense after the break, but they came close to stealing it late on when Keane powered a header just over the bar from a Sigurdsson corner. Azpilicueta then did brilliantly to rob Ramirez as the home side broke with numbers in their favour.
In the end, the sides couldn't be separated, so we take the point and now look forward to the hosting of Brighton at the Bridge on Boxing Day.
Everton (4-3-3): Pickford; Kenny, Jagielka (c), Keane, Martina; Schneiderlin, Gueye (Baningime 50), Davies (Williams h/t); Lennon (Ramirez h/t), Calvert-Lewin, Sigurdsson.
Unused subs: Robles, Holgate, Bolasie, Niasse
Booked Calvert-Lewin 10, Martina 21, Keane 56
Referee Bobby Madley