The Bangalore top order batsmen, except for Rahul (68 not out), failed to click after being put into bat but a late flurry of lusty blows from the wicketkeeper batsman saw the home side reach to a fighting total at Chinnasawmy Stadium.
Rahul, who came in when Bangalore were in a precarious 17 for two in the fourth over, struck three fours and four sixes in his 53-ball unbeaten innings.
He shared 43 runs with AB de Villiers (24) for the third wicket and another 53 with Sachin Baby (25 not out) for the unbroken fifth wicket as the home side went past 150-run mark on account of the late recovery.
Bangalore were 60 for 2 at the halfway stage but added 91 runs from the next 10 overs. Sachin gave an able support to Rahul towards the close of the innings with a quickfire knock of 13-ball 25 not out which had two fours and two sixes.
The cause of the home side was helped by the 23 runs Kieron Pollard leaked in the 18th over with Sachin Baby hitting two sixes and a four while Rahul also his share of a maximum hit.
For Mumbai, Mitchell McClenaghan, Tim Southee and Krunal Pandya took a wicket each.
Bangalore captain Virat Kohli (7), who has been in red hot form, met with a rare failure of the season. He struck a six in the fifth ball of the match bowled by Southee but was dismissed in the second over by Mitchell McClenaghan.
Kohli poked at a shortish delivery which was angling away, trying to guide it to third man, but only to find Harbhajan Singh taking an easy catch.
Two overs later, Chris Gayle (5) perished as he tried to smash a back of a length delivery on the off bowled by Southee with Rohit Sharma taking a well-judged catch.
AB de Villiers (24) and Shane Watson (15) got the starts but could not convert on them. De Villiers was out in the 11th over when he was shaping up for a big innings while Watson was run out as a result of a brilliant direct throw by Rohit.
RCB: V Kohli*c Harbhajan Singh b McClenaghan 7 C Gayle c Sharma b Southee 5 AB de Villiers c Rayudu b Pandya 24 KL Rahul† not out 68 S Watson run out (Sharma) 15 Sachin Baby not out 25 Extras (lb 5, w 2) 7 Total (4 wickets; 20 overs) 151 Fall of wickets 1-8 (Kohli, 1.1 ov), 2-17 (Gayle, 3.4 ov), 3-60 (de Villiers, 10.1 ov), 4-98 (Watson, 15.3 ov)