Simmons struck five sixes off 51 balls and Kieron Pollard weighed in with 41 off 17 deliveries to power the Indians up to 187-6 off their 20 overs.
Chennai struggled to 162 all out off 19 overs in reply – Faf du Plessis (45 off 34) lacking sufficient support from his fellow batsmen as Lasith Malinga (3-23) and Harbhajan Singh (2-26) chipped away.
The Super Kings could yet make it through to the final, however, if they overcome the winners of Wednesday’s Eliminator between Royal Challengers Bangalore v Rajasthan Royals.
In-form Mumbai – who finished second in the table despite losing their opening four matches of the competition – accumulated steadily after choosing to bat first as Simmons shared an opening stand of 90 in 10.4 overs with Parthiv Patel.
Simmons upped the rate after the first four overs yielded, smashing Ravichandran Ashwin for two sixes in three balls on his way to a 38-ball fifty.
Patel joined the party by taking 14 runs off the first three balls of the ninth over, bowled by Pawan Negi, before he holed out to long on for 35 off 25 deliveries.
Simmons followed him back shortly afterwards to make it 113-2 – Negi taking an excellent low catch out in the deep off the bowling of Ravindra Jadeja.
The platform was perfect for Pollard, who teed off to great effect despite seeing skipper Rohit Sharma sky a simple chance, the West Indian plundering five sixes.
His onslaught helped Mumbai chalk up 46 runs off the final three overs of the innings – Pollard falling three balls from the end as he sought to strike Bravo for a second six in as many deliveries.
Mumbai’s disappointment at not posting a total nearer 200 was tempered when Malinga trapped Dwayne Smith lbw with the fourth ball of Chennai’s chase, although replays showed that the ball would have missed leg stump.
Two runs came off the over but the next cost 17 as Faf du Plessis went on the offensive against Mitchell McClenagan before the South African received a lifeline two overs later when he had 14 as Malinga shelled a simple catch at third man.
Michael Hussey quickly raced to 16 but could not go on as he charged Vinay Kumar but only succeeded in snicking behind.
At the halfway point, Chennai were right in the game at 86-2 needing another 102 for victory but the match turned in the 11th over as Suresh Raina chipped the ball straight back to Harbhajan Singh to go for 25 off 20 balls.
Chennai skipper MS Dhoni went very next ball for a golden duck, trapped lbw by the spinner as he attempted to sweep to make it 86-4. Harbhajan spurned the chance to claim a third wicket in the over when he shelled a sharp return catch offered by Bravo on just one.
With the game slipping away, Du Plessis holed out off Jagadeesha Suchith and Chennai’s fate was all but sealed when Bravo was run out at the non-striker’s end as he dawdled over a second run to make it 119-5.
Ashwin gave it a go with some lusty blows yielding 23 off 12 balls but it was too little, too late as Mumbai qualified for their third IPL final.