Returning fast bowler Ishant Sharma, playing his first IPL match since 2021, returned figures of 2-19 to help bowl out Kolkata for 127 in New Delhi.
But Delhi made tough work of their modest chase and despite Warner's fourth half-century of the season achieved their target with four balls to spare in a rain-delayed match that went past midnight.
Delhi, who are still hunting for their maiden IPL title, remain bottom of the 10-team table with just one win in six matches.
"I was thinking it was like my first Test - happy to get off the mark," said Delhi's director of cricket and former India skipper Sourav Ganguly.
"We have bowled well this season. These are good pitches and fast outfields, but the problem is the batting. We need to go back and look at ourselves as to how we can do better."
The left-handed Warner started strongly with a few boundaries but Delhi stuttered after they lost Prithvi Shaw, Mitchell Marsh and Phil Salt in quick succession.
Leg-spinner Varun Chakravarthy sent back Warner lbw for his second wicket and Manish Pandey became impact player Anukul Roy's second scalp to raise Kolkata's hopes.
Axar Patel (19) and Lalit Yadav (4) stayed calm to steer Delhi home.
"Credit to the bowlers that if we can fight with this total then the coming matches should be good," Kolkata skipper Nitish Rana said after his team's third straight loss.
"They batted well in the powerplay and we came back well, but we were 15-20 runs short."
The second match of the day started an hour late due to evening rains in the national capital and the home team elected to field before picking up an early wicket in Bangladesh's Liton Das.
Ishant stood out with key wickets of Rana and Sunil Narine before being substituted under the new impact player rule.
Others joined in as pace bowler Anrich Nortje, spinners Axar Patel and Kuldeep Yadav took two wickets each to bundle out the opposition on the final ball of the innings.
England opener Jason Roy hit a valiant 43 in his first outing this season and Andre Russell overcame a slow start to smack an unbeaten 38 with three successive sixes in the final over.
