RCB, opting to bat, put up 162 for eight with Ab de Villiers hammering 43 off 27 balls.
It was a rather straightforward chase for Mumbai as the home team reeled off 59 runs in the last six overs, courtesy some crisp hitting from Rohit, who ended with 56 off 37 balls.
Hardik Pandya remained unbeaten with the captain, scoring 14 off nine balls.
With the win at Wankhede, Mumbai moved to the top of the points table with eight wins out of 10 games fetching them 16 points.
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Fittingly, Rohit hit the winning runs, a cracking drive past mid-off. It was one of the six boundaries he hit alongside a maximum.
The quick start to the chase was provided by Jos Buttler (33 off 21) and Nitish Rana (27 off 28) after Parthiv Patel was out for a first ball duck.
Earlier, Mitchell McClenaghan claimed three wickets as Mumbai Indians produced a disciplined bowling performance to restrict RCB to a modest total.
For RCB, de Villiers emerged as the top-scorer after he blasted three fours and as many sixes during his 27-ball innings, while Pawan Negi (35) and Kedar Jadhav (28) provided the late charge to take the team past the 150-mark.
RCB lost their first wicket in the fourth over when Mandeep Singh (17) went for the slog sweep and was caught by Hardik.
In the sixth over, McClenaghan removed Virat Kohli (20) when the RCB skipper flicked a slower ball straight to short midwicket and Rohit took a simple catch, reducing the visitors slipped to 40 for two.
The South African gave Krunal Pandya the same treatment as RCB accumulated 13 runs off the ninth over.
After 10 overs, they were 80 for 2 but Travis Head (12) holed out to Hardik at deep midwicket and RCB slumped to 85-3.
De Villiers brought up the 100 with a six off Krunal over the square leg boundary but was dismissed in the next delivery by the same bowler for the fourth time in as many meetings.
Negi and Jadhav then steadied the innings, taking ones and twos to keep the scoreboard ticking.
Negi finally exploded in the 18th over, producing a couple of sixes off Lasith Malinga over extra cover and long- off, before smashing a six over long-on boundary in the next over to bring up the 150.
The final over yielded just five runs as McClenaghan dismissed Negi and Jadhav in successive deliveries, while S Aravind ran himself out while looking for a single.