With this victory, MI have dethroned Sunrisers Hyderabad to acquire the third position on the points table with 8 points in their kitty.
Pollard anchored the MI innings with his 31-ball knock punishing the Punjab bowlers with 10 sixes and three fours. The burly West Indian was dismissed in the second ball of the last over with four needed for victory from final five balls.
Alzarri Joseph (15 not out), who added a crucial 54 runs in 23 balls with Pollard for the seventh wicket, and Rahul Chahar (1) managed to hold their nerves to take side past the finish line.
After 10 overs, MI needed 133 in 60 balls, still fancying their chances with the big-hitting Pollard at the crease and another fine finisher Hardik Pandya to follow.
Shami got rid of the other Hardik and Krunal Pandya in space of four balls.
The equation whittled down to 63 from 30 balls and then 15 off the last six balls after Pollard smacked Curran for two sixes and a four in the penultimate over before he was caught off Rajput in the last over.
Earlier in the evening, Rahul smashed his maiden IPL hundred while Chris Gayle struck a 36-ball 63.
Gayle, who smashed seven towering sixes and also hit three fours, laid the platform for the big total with his opening partner Rahul, who carried his bat through to remain unconquered on 100, laced with six sixes and as many fours.
The duo's 116-run stand in 77 balls took MI by storm. Gayle's dismissal in the 13th over slowed down the innings but Rahul punished Hardik Pandya by hitting three sixes and a four in the 19th, which yielded them 25 runs.
Gayle's mayhem ended when he was caught at deep mid-wicket off Behrendorff to nip his opening partnership with Rahul. The left-handed Gayle's departure slowed down the run-rate with Hardik sending back David Miller and Karun Nair cheaply and Bumrah accounting for Sam Curran between the 15th and 18th overs.