Ruturaj Gaikwad scored 88no from 58 balls to set up CSK’s total of 156-6, with the reigning champions falling short in their pursuit.
Chennai had been in big danger early, slumping to 24-4 after captain MS Dhoni was dismissed for 3 but opener Gaikwad led the revival, with help from Ravindra Jadeja (26 from 33) and Dwayne Bravo (23 from 8).
Bravo also took 25-3 from his four overs after Deepak Chahar (19-2) had dismissed both openers cheaply.
Mumbai’s run chase never got going with Quinton de Kock out LBW to Chahar for 17 and Suryakumar Yadav only managing 3 from 7 balls.
Saurabh Tiwary provided the strongest resistance with an unbeaten 50no for 40 balls, but wickets kept falling.
Dhoni said: “At 30-4, you want to put up a respectable score, I felt Rutu and Bravo got us more than what we expected. We thought of 140, to get close to 160 was tremendous.
“The wicket was two-paced, slightly slow to start off. Also it’s difficult if you’re batting lower down, if you don’t get a chance to see what is happening, you want to go hard.
“We batted sensibly and finished superbly. It was sensible for one batter to bat right through to the end.”
Defeated skipper Kieron Pollard hailed Gaikwad for his decisive role, with the Indians unable to find the runs late after needing 10 an over with six overs left and six wickets in hand.
“Ruturaj batted really well,” Pollard said. “In T20 cricket, if one person bats through he can hurt you. We didn’t finish off well with the ball.”