Starc dismantled the top order of Sunrisers' normally formidable batting line-up, dismissing Travis Head among a trio of early wickets in Visakhapatnam.
He then returned to mop up the tail, dismissing Sunrisers for 163, and claiming his first five-wicket haul in the Indian Premier League.
Jake Fraser-McGurk made 38 off 32 balls as Delhi chased down the target with four overs to spare.
Starc now has eight wickets in two matches for his new team, having joined them after being released by Kolkata Knight Riders despite being man of the match when taking 2-14 for KKR in last year's final against Sunrisers.
"It's been a good day," said Starc. "All round, we were fantastic. I've played for long enough, so I know what to do. You've got to think outside the box, even as an older player you have to bowl different balls.
Now 35 Starc said he had no time-frame for quitting T20s. "I'm still enjoying being competitive, that's why I'm still playing," he added.
Hyderabad opted to bat first but lost Abhishek Sharma run out for one after a mix-up with Head. Then Starc stepped up and reduced them to 4-37 in 4.1 overs.
Ishan Kishan, a centurion in the previous game, was caught at deep backward point for two, Nitish Reddy dismissed for a two-ball duck, and Head caught behind for 22 off 12 balls. It was the sixth time Starc had dismissed Head in eight T20 innings at a cost of 34 runs.
"I think that's why he doesn't face the first ball any more," said Starc, half-joking.
Aniket Verma (74 off 41 with six sixes) and Heinrich Klaasen (32 off 19) rebuilt the innings but, aided by a superb catch fro Fraser-McGurk, Kuldeep Yadav restored Delhi's control and Starc finished Sunrisers off in the 19th over.
In reply Fraser-McGurk and Faf du Plessis (50 off 27) put on 81 in nine overs and though Zeehan Ansari (3-42) dismissed both in the tenth Delhi cruised to victory . Cummins' two overs went for 27.
