Chasing a stiff target of 166, Mumbai had a flying start with opener Dwayne Smith (62) once again coming good up front, before Rajasthan Royals bounced back with Kevon Cooper (2/33) striking in the middle.
Mumbai Indians almost choked in the middle yet again but some sensible batting by young Rishi Dhawan and Harbhajan Singh helped them pip Rajasthan.
It will thus be a repeat of the 2010 final which Mumbai lost.
But Pollard calmed the nerves with a first-ball six taking them closer to the target.
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There was more drama in store when James Faulkner dismissed Pollard in the next over with 25 runs needed from 15 balls and with five wickets in hand.
With 23 needed from the last two overs, Rayudu hit Cooper for a six before being dropped by Brad Hodge, which proved to be the turning point of the game.
With six needed from four, young Rishi Dhawan, who was included in place of Munaf Patel, showed amazing calm in pressure situation as he walked across the crease before hitting a boundary with a scoop shot that almost settled the issue for Mumbai.
A target of 166 might look stiff from the Eden Gardens' context but Mumbai Indians approached it in a quiet yet effective manner with Tare and Smith giving them a flying start.