Pollard, who hit three fours and five sixes in his aggressive knock, shared 93 runs for the sixth wicket to set up Mumbai's comfortable win at the Chinnaswamy Stadium.
Pandya hit a six off Tymal Mills as Mumbai chased down the target of 143 by reaching to 145 for 6 with seven balls to spare.
This was Mumbai's third win on the trot after losing their campaign opener against Rising Pune Supergiant.
RCB failed to capitalise on the good work done by Kohli, managing just 31 runs off the last five overs and lost four wickets.
Kohli, out of action for almost a month with a shoulder injury he suffered during the third Test against Australia, opened the innings and he got into the groove in no time.
After mistiming a short ball off Harbhajan Singh for a single in the second over of the innings, Kohli found his ominous touch.
However, RCB lost their way after Kohli's dismissal in the 16th over, struggling to 127 for five from 110 for one in a space of 17 balls.
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