Stoinis returned brilliant figures of 4/15 in his full quota of four overs after Mumbai Indians skipper Rohit Sharma elected to bat.
Mumbai Indians were off to a disastrous start, losing two wickets for just eight runs on the board within the third over of the innings.
Playing in place of Parthiv Patel, Unmukt Chand could not make the rare opportunity count as he got out to a terrible shot, giving an easy catch at mid-on off the bowling of Mohit Sharma while trying to whip a length ball into the leg side.
Mumbai Indians were dealt a body blow when they lost the in-form Rohit, bowled by left-arm spinner Axar Patel. Rohit made 15 off 24 balls with a lone boundary.
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The reigning champions sank deeper as they lost Nitish Rana (25) followed by the big wicket of Jos Buttler (9). Stoinis got both the wickets to leave Mumbai reeling at 61 for five in the 14th over. Rana struck three sixes while facing 28 balls.
Pollard was well supported by Krunal Pandya as he also, on and off, found the fence before holing out at the deep to end a sixth-wicket partnership of 42 that came off just 19 balls.
Stoinis then returned to dismantle Pollard's off stump and again pegged MI back at a crucial juncture. While Pollard smashed 27 off 20 balls with the help of three sixes, Pandya contributed a 12-ball 19, hitting a boundary and six.
Stoinis was supported by Sandeep Sharma (2/11) and Mohit (2/26).