Chasing 191 for a win, Mumbai faltered against Delhi's smart catching and great spin force, ably led by Tahir (3-22), to manage only 153 for nine in 20 overs.
Earlier, big fifties from Duminy (78 not out) and Shreyas Iyer (83) helped DD set an imposing target of 190 for four.
Mumbai Opener Lendl Simmons (15) let go of a quick start to mis-hit a Domnic Muthuswami delivery as Duminy pulled off a sensational running catch.
The Delhi catching continued to impress the home crowd as Tahir then took a brilliant outfield catch to send Parthiv Patel (28) back in the dug-out off Angelo Mathews.
Patel, who had hit 4 fours and a six in his 19-ball blitz, pulled the ball over mid-wicket only to see Tahir cover good ground and take a sharp catch at deep square leg.
DD bowlers kept chipping in with wickets as leg-spinner Amit Mishra (2-32) got Delhi boy Unmukt Chand (14) stumped to make the MI task difficult.
Captain Rohit Sharma (30) and Ambati Rayudu (30) then tried to get the chase on track, stitching a 49-run fifth wicket partnership.
Wicketkeeper Kedhar Jadhav though did miss a stumping off Mishra to give Rayudu a life in the 13th over. Rayudu was on 5.
But the DD fielders continued to take their catches as Nadeem took another neat skier at sweeper cover off Nathan Coulter-Nile to send dangerman Rohit back in the hut.
Hardik Pandya (0) could hardly blink as Tahir got him caught at long-off. On song in the 17th over, Tahir struck again to get Rayudu caught at long-off once again.
Lasith Malinga played out the hat-trick ball but Tahir firmly got the purple cap on his head.
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