The Super Kings won the toss and chose to bowl, but couldn't take an early wicket as openers Lendl Simmons (67) and Mike Hussey (39) put on a 77-run partnership off just 58 balls to start the hosts' innings.
Double-figure scores were added by Corey Anderson (20), Rohit Sharma (20) and Kieron Pollard (14), but none could really get going and the Indians' tail-end failed to score well at the end as they finished on 173-8.
Although one of the more expensive bowlers, Mohit Sharma led the Super Kings' wicket-taking with three for 42 runs.
Two-time champions Chennai had to chase down 174 to avoid elimination from this year's competition and they managed to get there with eight balls to spare.
Mumbai could only take three wickets as Suresh Raina (54 not out off 33 balls) led the charge for Chennai.
Two of the wickets came from the bowling of Harbhajan Singh, who finished with impressive figures of 2-27.
Raina was joined by David Hussey for the fifth wicket and they put on an unbeaten 89 off just 55 balls to see the Super Kings home.
Chennai now face Kings XI Punjab, who lost to the Kolkata Knight Riders earlier today in the first semi-final, in Friday's second semi-final, with the winner going on to face Kolkata in Sunday's final.