Ambati Rayudu (34no from 19 balls) and Hardik Pandya (21no from eight) blasted the visitors home with four balls to spare and above Royal Challangers Bangalore into the fourth and final play-off berth.
Chennai, who were playing in their record 150th T20 match, had posted 158-5, with MS Dhoni (39 from 32) and Pawan Negi (36no from 17) cracking sixes late on.
But the home side had no answer to Rayudu and Pandya at the backend of the Mumbai knock as they sunk to their first loss at M. A Chidambaram Stadium since 2013.
Brendon McCullum (23 from 11) creamed Vinay Kumar for three successive fours in the fifth over of Chennai???s innings before being caught at deep-midwicket by Pandya next ball failing to nail a short delivery.
Suresh Raina (10) and Dwayne Smith (27) picked out Pandya, too, as Super Kings stumbled to 62-3 at the halfway stage, but Dhoni and Negi then plundered a combined six boundaries and three sixes.
Spinner Jagadeesha Suchith (1-21 from fours overs) was the pick of Mumbai???s bowlers, the 2013 IPL champions unable to play Sri Lanka???s Lasith Malinga in Chennai on political grounds.
Mumbai openers Parthiv Patel (45 from 32) and Lendl Simmons (38 from 31) got their team off to a flyer, with the visitors going in excess of 10 runs an over from the first six, but the scoring dried up and both batsmen were dismissed by fit-again Ravichandran Ashwin in the 11th over.
Kieron Pollard (1) was run out by fellow West Indian Dwayne Bravo in the next over and when captain Rohit Sharma (18) perished from the first ball of the 18th with Mumbai still requiring 34, Chennai looked favourites.
But, amid some shoddy Super Kings catching, Rayudu and Pandya smoked the maximums ??? the 19th over, bowled by Negi, going for 25 runs ??? to keep Mumbai???s stellar run going.