After the hosts had made 165-7 in Pune, Delhi appeared to be cruising to victory while man of the match Mayank Agarwal (68) and Yuvraj Singh (55) were adding 106 for the third wicket.
However, that duo fell on the same score and the Daredevils ended up getting home with just one delivery to spare.
It was a very welcome victory for JP Duminy???s men, who had suffered last-ball defeats in both their first two matches.
Kings XI were looking to make it two wins from three and they chose to bat first after winning the toss.
Opener Virender Sehwag top-scored for the hosts with 47 from 41 balls, and added 71 for the second wicket with keeper Wriddhiman Saha (39).
Glenn Maxwell then produced a big-hitting cameo, taking 15 off his first four balls, including two sixes, before holing out to his fifth delivery.
He was dismissed by Imran Tahir and the spinner also removed captain George Bailey (19 off 11), although he was also quite expensive, finishing with 3-43 from four overs.
Delhi lost Shreyas Iyer (6) early on as they began their hunt for 166 and Duminy (21) was run out in quite bizarre fashion when bowler Axar Patel deflected an Agarwal drive onto the stumps and collided with Duminy in the process, replays showing the batsman was unable to get back into his ground before the bails came off.
Delhi shrugged off the misfortune, Agarwal and Yuvraj combining in a fine partnership which took them to within seven runs of victory.
However, Singh was caught off Anureet Singh in the penultimate over and Agarwal was bowled by the very next delivery.
Kedar Jadhav (3) then fell with the scores level in the final over but Angelo Mathews (six not out) eventually hit Patel straight for four to wrap up the win with a ball to spare.