Pollard hit a timely unbeaten half-century to lead Mumbai Indians to a competitive 148 for six after the visitors elected to bat at M A Chidambaram stadium.
Chennai skipper M S Dhoni almost won the match for his side with a breathtaking 51-run knock off 26 balls but his dismissal in the first ball of the 20th over scripted Mumbai's sensational win.
With 12 runs to win from the last over, Dhoni hoicked Munaf Patel powerfully but ended up caught by an air-borne Pollard at the boundary ropes. That stunning catch sealed the fate of Chennai.
The hosts managed 139 for nine as they paid price for losing wickets intermittently, not building partnerships.
It was all due to Dhoni's heroic knock, which was studded with five fours and three sixes, that Chennai could come close to the target since the hosts were reeling at 99 for six in 16 overs at one stage.
Mumbai were struggling at 59 for four in the ninth over when Pollard came to the crease and the West Indian bailed his side out of troubled waters with his 38-ball knock, which was laced with nine boundaries. The burly Caribbean ended the Mumbai innings with his fifth six.
A charged up Mumbai began the defence of their total impressively as the bowlers troubled the Chennai batsmen by keeping a tight line even as fielders dropped a few chances.
Murali Vijay (5), Suresh Raina (10) and Michael Hussey (20), who was dropped twice, were all back in the pavilion by the eight over.
Such was the domination of the Mumbai bowlers - pacers spinners alike - that in the first 10 overs Chennai batsmen could manage just five boundaries, three of them coming from the bat of Hussey.