Kagiso Rabada (3/37) and Anrich Nortje (2/12) were the pick of the bowlers as Sunrisers were restricted to 134 for nine.
Shreyas Iyer hit 47 as Delhi reached the target with 13 balls to spare.
The win takes Delhi to the top of the table, overtaking Chennai Super Kings (CSK) but having played a game more while Sunrisers remain bottom.
David Warner’s return to the Sunrisers’ starting XI did not go as planned with the Australian dismissed for a three-ball duck by Nortje.
Warner was replaced as Sunrisers’ Captained by Kane Williamson after the first six matches of the competition, and then dropped for the seventh game, before the tournament was postponed.
But his tame dismissal – a leading edge chipped to point – could now mean an opportunity for England opener Jason Roy in Sunrisers’ next fixture against Punjab Kings on Saturday.
Delhi Capitals: 139/2 in 20 overs (Shreyas Iyer 47 n.o, Shikhar Dhawan 42)
Sunrisers Hyderabad: 134/9 in 20 overs (Abdul Samad 28, Rashid Khan 22, Kagiso Rabada 3/37). (BBC)