Nair amassed 89 off 40 balls until he was bowled out by Mitchell Santner leading DC to lose steam in the chase for 206.
He hit 12 fours and 5 sixes fuelling the home team's cause.
After Nair walked back to the hut with ample opportunity to cross MI's 205-5, DC lacked the necessary support from the middle order leading the home team to get bowled out for 195 with one over to spare.
Karn Sharma picked three wickets for 36 in four overs for the visitors who clinched a 12-run victory.
Earlier, Tilak Varma (59 off 33 balls), Suryakumar Yadav (40 off 28 balls) and Ryan Rickelton (41 off 25 balls) powered MI to 205-5 in 20 overs.