The hosts overcame loss of two wickets in the first over to post a challenging 186 for eight after being invited to bat and skittled out the visitors for 159 with nine balls to spare at the Feroz Shah Kotla ground.
Thanks to fighting half-centuries by Nair (68 off 50 balls) and Billings (54 off 34 balls) combined with Carlos Brathwaite's (34 off 11) finishing touches, Daredevils' innings shaped well as they registered their best total against KKR.
It was first defeat for KKR in this tournament while chasing as except opener Robin Uthappa (72 off 52) no other batsman showed resistance.
Both Nair and Billings impressed with their batting as they desisted from swinging the bat wildly, playing proper cricketing shots. They were comfortable against both pacers and spinners, building the innings with smart cricket.
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Braithwaite played true to his big-hitting reputation by smashing 34 with three sixes and as many fours to ensure that the hosts end with a bang.
Andre Russell was the most impressive KKR bowler with figures of 4-0-26-3 while Umesh Yadav (3\33) also picked three wickets.
Zaheer Khan did exactly what he was expected to do after not-so-great performance in the previous matches by bowling with a lot of venom and taking two early wickets.
He had rival skipper Gautam Gambhir (6) caught off a leading edge at covers and trapped Piyush Chawla (8), who was promoted up the order.
Opener Robin Uthappa was keeping one end tight but Yusuf Pathan (10) started his walk back to dug out when he handed an easy catch to Amit Mishra off Braithwaite.
(REOPENS DEL 34)
Suryakumar's knock ended when he failed to connect one from Brathwaite and was taken splendidly by Shreyas Iyer.
Uthappa got to his fifty with a six off Mishra but had lost R Sathish by then. He hit another four off Mishra as KKR now needed 60 off last five overs.
Earlier, Russell rattled the host camp by removing both the Delhi openers - Shreyas Iyer (0) and Quinton De Kock (1) -- in the first over of the match.
Iyer was caught plumb in front of the wicket and De Kock miscued a pull that landed safely in the hands of Brad Hogg.
It was now left to Nair and Sanju Samson to build the innings. Both the young batsman showed good composure, playing brilliantly with straight bat. Their shots were proper cricketing strokes. They drove effortlessly and were elegant when played through the leg side.
Nair kept going strong, flicking Russell for a four when bowled up and reverse swept Narine. The young English batsman, Billings also made an impression as he pulled a short one from Umesh Yadav, who bowled at a good speed.
Nair consistently rotated the stroke and found odd boundaries as the hosts managed 70 runs on the board at half-way stage.
As Billings grew in confidence, he too started to play his shots. And he too played proper shots instead of swinging his bat wildly at everything and anything. His cover drives off Hogg was a treat to watch.
Billings too pressed the accelerator button, hitting Holder for the first six of the innings. It was pitched outside off but Billings went down and picked and swept it over square leg.
Nair lofted Umesh Yadav elegantly for a six over long-off but was trapped by the paceman in the next ball.
It brought Chris Morris to the crease but the South African could not repeat his last-match heroics and was bowled for a duck by Yadav.
Billings found his fifty with a six off Hogg but was dismissed by Yadav, castled on a slower full toss. However he had the done job for his side by then.