Mumbai batted first and posted a total of 148 for six on a difficult wicket as both sides struggled, and Chennai fell just short with 139 for nine in response.
The visitors found themselves under the cosh with 83 for six on the board, but Pollard proved the difference with his explosive unbeaten knock.
Mahendra Singh Dhoni almost grabbed an unlikely victory for the Super Kings with a brisk 51 in reply, but it proved to be insufficient.
In the other match of the day, the Rajasthan Royals secured a five-run victory over the Delhi Daredevils in their match in Delhi.
Rajasthan put 165 for seven on the board with Rahul Dravid scoring 65, only for the destructive David Warner to scare the visitors with a quickfire 77 with his side hurtling to victory.
But Kevon Cooper intervened with a fine over and acting captain Brad Hodge produced an inspired run-out to dismiss Warner in the penultimate over.
Nine runs were required off the final over, but Cooper managed to ensure a fine victory for his side with Naman Ojha unable to hit the required maximum off the final delivery.
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- Rajasthan Royals