Rohit, who remained unbeaten on a 35-ball 59, combined with the burly West Indian to forge a 60-ball 97-run partnership for the fifth wicket at the Wankhede Stadium.
Rohit's 68-minute innings yielded three fours and four sixes, while Pollard struck six boundaries.
Put into bat, Mumbai started well but lost Aussie opener Ben Dunk (15) in the first ball of the fourth over. He shared a 25-run stand with wicketkeeper batsman CM Gautam and had struck a couple of fours against Ashoke Dinda before paying the price for a loose shot that was caught by Yuvraj Singh off Harshal Patel.
Gautam departed in the next over. He struck three sixes and two fours in his 28-ball 30.
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Corey Anderson, who hit a six over long on off leg-spinner Yuzvendra Chahal, perished while attempting the same shot two balls later, reducing the hosts to 84 for four in the tenth over.
It was upto skipper Rohit and Pollard to do the repair job. Rohit who got off to a slow start, accelerated and thrashed pacer Varun Aaron in the penultimate over for 24 runs. He carted the bowler for three sixes and a four to bring up his half century.
The pacer who had bowled a bouncer earlier, said something to the batsman, who dismissed it with a wave. In the next ball, even though Pollard pulled out of the strike, Starc fired the ball at the batsman, angering the Trinidadian, who appeared to fling the bat at the bowler's end but, luckily, it slipped out of his hand.
Pollard was run out following a misunderstanding in the last over. Royal Challengers Bangalore conceded 48 runs in the last three overs. Dinda was the most profligate and gave away 38 runs in his three overs, while Aaron was expensive as he conceded 41 runs in his four.