Young Sanju Samson (60) struck a fine half-century while JP Duminy made a brisk unbeaten 49 to help Daredevils post 164 for five after being asked to bat at the Feroz Shah Kotla ground.
Defending the total, Daredevils dished out a disciplined bowling and fielding display to restrict Mumbai Indians to 154 for seven and register their third win on the trot.
Veteran Amit Mishra proved why he is regarded as the best leg-spinner of the country at present by returning with figures of 4-0-24-2, which included the important wickets of Ambati Rayudu and Jos Buttler.
While Rayudu played an aggressive knock, Rohit's was an uncharacteristic innings as he preferred to play the second fiddle. Rayudu cracked four boundaries before he was cleaned up by a flighted googly from Mishra in the ninth over.
After Rayudu's fall, Krunal Pandya came to bat and the youngster used his long handle to great effect to make a quickfire 36 off just 17 balls before he was caught short off the crease at the non-strikers end by a direct hit from Daredevils skipper Zaheer Khan off his own bowling.
The way Krunal was going at one stage it looked Mumbai would finish off the match in quick time but his run out and Mishra's brilliant spell in the middle overs turned the things in Daredevils favour. Mishra then foxed Butler with another googly to trap the batsman plumb in front of the wicket.
With Mumbai Indians needing 47 off the last four overs for a win, all hopes of the visitors were on the shoulders of their skipper Rohit and Kieron Pollard.
But it was not to be Mumbai's day as Pollard departed in the next ball when he miscued a shot only to give a simple catch to Chris Morris at sweeper cover.
Needing 21 off the last over, Rohit brought down the equation with a six off the second ball of Morris but he fell victim of an unnecessary mix up with Hardik Pandya in the very next ball while going for a double.
Earlier, Samson top-scored with 60 off 48 balls during which he struck four boundaries and two sixes, Duminy's runs came off just 31 balls with the help of three fours and two hits over the fence.
However, the start was far from rosy for the Daredevils as they lost IPL 9's maiden centurion and in-form Quinton de Kock (9) early.
De Kock started off Derdevils innings in great fashion by hitting Tim Southee to the fence in the first two balls of the innings. But the South African departed in the next over, caught by Hardik Pandya at backward point off Mitchell McClenaghan while the batsman went for an expansive shot.
To make matters worse for the hosts, Karun Nair too did not last long as he top-edged a Harbhajan Singh delivery to Southee a fine leg as Daredevils slumped to 54 for three by the end of the eighth over.
Then Duminy and Sanson joined hands and played sensibly to stitch vital 71 runs off just 51 balls for the fourth wicket to take Daredevils forward.
The duo mixed caution with aggression to keep Daredevils scoreboard ticking and set the platform for the total.
After Samson's dismissal Duminy took the onus on himself to take Daredevils to a decent score and in the company of Pawan Negi (10 not out) put on 39 runs for fifth wicket to take the hosts past the 160-run mark.
For Mumbai Indians McClenaghan (2/31) picked up two wickets, while Hardik Pandya (1/7) and Harbhajan Singh (1/24) accounted for one wicket each.
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