Ambati Rayudu hit an unbeaten 81 and Kieron Pollard remained undefeated on 52 as Mumbai made 173-5 with two balls to spare after Royal Challengers put up a competitive target of 171-6.
Bravo (11) lofted seamer Rajat Bhatia's full toss over long on at Eden Gardens as Chennai Super Kings made 160-5 after Knight Riders had posted 158-6 off their 20 overs.
Both Rayudu and Pollard shared a match-winning 122 runs for the sixth wicket stand off 65 deliveries after Mumbai Indians lost star batsman Sachin Tendulkar for naught and struggling on 51-5.
Rayudu's knock came off 54 balls and it featured six fours and four sixes. Pollard, who hit five fours and three sixes, raised his half century when he hit countryman Chris Gayle for a winning boundary in the last over.
Needing 14 to win, Pollard flicked off-spinner Gayle over the midwicket fence for a six before hitting two successive boundaries.
Earlier, Mayank Agarwal hit six boundaries and four sixes in scoring an unbeaten 64 off only 30 balls to lift the Royals total, hitting seamer Munaf Patel for 24 runs in the last over of the innings.
Mumbai Indians now have 18 points from 14 matches. They share top spot with Delhi Daredevils, who are on the same points, but have a game in hand on Mumbai.
Royal Challengers are in fifth place in the nine-team league with 15 points after winning seven of its 14 matches.
Super Kings' opening pair of Michael Hussey (56) and Murali Vijay (36) put on 97 runs before off-spinner Sunil Narine put the brakes on and removed both batsmen in the space of three balls.
Super Kings needed five off Bhatia's last ball of the match and Bravo smashed the full toss for six to take his team to fourth place with 17 points.
Earlier, Bhatia had featured in the three dismissals when he first had Suresh Raina run out and then had the wickets of Faf du Plessis (13) and also had Mahendra Singh Dhoni (28) clean bowled in the last over.
Knight Riders could not capitalise on a fine start provided by their captain Gautam Gambhir (62) and Brendon McCullum (37), who shared a first wicket stand of 99 off 69 balls.
Knight Riders are also on 17 points but are third because of a better run-rate than Super Kings.
Brief scores:
Mumbai Indians 173-5 in 19.4 overs (Ambati Rayudu 81 not out, Kieron Pollard 52 not out; Muttiah Muralitharan 1-16) beat Royal Challengers Bangalore 171-6 in 20 overs (Mayank Agarwal 64 not out, Tillakaratne Dilshan 47; Kieron Pollard 1-20, RP Singh 1-23).
Chennai Super Kings 160-5 in 20 overs (Michael Hussey 56, Murali Vjay 36; Sunil Narine 2-14, Rajat Bhatia 2-33) beat Kolkata Knight Riders 158-6 in 20 overs (Gautam Gambhir 62, Brendon McCullum 37, Shadab Jakati 2-26).
