Despite the poor state of the pitch, Musa got his first chance of scoring on 19 minutes but fired off target from close range.
Two minutes later, the Nigeria captain opened scoring. He smashed in a left-footed effort from a corner kick past Kuban goalkeeper Aleksandr Belenov.
Playing as a centre forward with the trio of Zoran Tosic, Aleksandr Golovin and Roman Eremenko behind him, Musa's pace troubled Kuban and he came close to scoring again on 27 minutes but Belenov was in the right place to make a save.
Seven minutes into the second half, Eremenko did well to pick out Musa, whose shot from outside the box was blocked by Kuban's defence.
Within seconds Musa was on the prowl again but this time his left-footed shot just missed target by inches.
Just before the hour, Musa turned provider for Eremenko but the Russian-born Finnish forward saw his effort blocked.
Musa would not be denied his second goal of the game with nine minutes to time when he finished off Alan Dzagoev's assist into the centre of the post for the insurance goal.
The win for CSKA put them two points clear at the top of the Russian league as Musa now has nine league goals.
The Nigeria forward is third leading scorer in Russia behind Spartak Moscow's Quincy Promes and Zenit's Artem Dzyuba who have 12 and nine goals respectively.
Musa is in third by virtue of having one assist less than Dzyuba who has three assists despite both men being tied on same number of goals.