Electing to bat, Chennai scored just 119 for 6 with Zaheer returning with superb figures of 2 for 9 from his four overs and Albie Morkel chipping in with two wickets as Delhi produced a disciplined bowling show at the Shaheed Veer Narayan Singh International Stadium.
Chasing the small target of 120, Delhi, who are already out of contention for a play-off berth, lost two quick wickets to be reduced to 24 for two in the fifth over as opener Quinton de Kock (3) and captain Jean-Paul Duminy (6) were out cheaply.
20-year-old Iyer, who was dropped on 32 in the ninth over, hit 10 fours and a six in his 49-ball unbeaten knock on a slowish track while Yuvraj, who was out in the 14th over, struck four boundaries and a six in his 28-ball innings.
Iyer hit the winning run, a single, as Delhi reached 120 for 4 in 16.4 overs.
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For Chennai, Ishwar Pandey and Pawan Negi took two wickets apiece.
Interestingly, Delhi snapped their seven-match losing streak against Chennai in the IPL.
For Chennai, today's loss means they are yet to seal a play-off berth though they still remain on top of league table with 16 points from 13 matches with one match left to play.