Yuvraj Singh scored a timely 70, his second half-century of the season, and raised a 93-run stand with Moises Henriques (25) to lead the visitors to a challenging 185 for three after being asked to bat.
The hosts, who have been enduring a tough season with one defeat after another, then came up with an inspiring collective bating show to stun the defending champions with five balls to spare.
Each and every Delhi batsman, who took strike, contributed in the win. The openers Karun Nair (39) and Sanju Samson (24) scored 40 runs in first four overs to provide a decent start and the batsmen who followed them built on the momentum.
Nair was dropped on 20 by Bhuvneshwar Kumar but the Daredevils stand-in captain could not make full use of the life and was dismissed by Siddarth Kaul.
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Runs came at a good pace for the hosts with young batsmen batting sensibly. They needed 61 from last seven overs and with Anderson and Iyer at the crease, it was not a big task. However, Iyer was dismissed by Bhuvneshwar.
Since the Delhi batsmen lost wickets after getting their eyes in, it kept Sunrisers' hopes alive but gradually the required run rate dropped and Corey Anderson (41 not out) finished the match in company of Chris Morris (15 not out).
Incidentally, Daredevils' last win had come at this very ground against Kings XI Punjab.
SRH were off to a flying to start through skipper David Warner (30) and Shikhar Dhawan (28) but Delhi spin duo of Amit Mishra and Jayant Yadav managed to choke the defending champions in the middle overs.
Sanju Samson dropped Yuvraj on 29 and the dashing left-hander exploited it to get some crucial runs for his side. After making 62 against RCB in his first match, he was struggling for runs and found touch when the team needed.
In the last three overs, Sunrisers collected 52 runs with Kagiso Rabada leaking 39 runs in 18th and 20th overs.
Earlier, Mishra played a crucial role in choking the Hyderabad batsmen as he gave away only 23 runs in his four overs and also got rid of Dhawan.
Jayant Yadav, included in place of Shahbaz Nadeem, also did a great job by conceding only 26 runs in his four overs.
Warner began from where he left against KKR, striking the ball confidently. He found boundaries off Yadav and Rabada with remarkable ease.
However, Shami took revenge in the next ball with a well- placed yorker that uprooted Warner's off stump.
Mishra has not bowled during Power Play overs this season and the trend continued. He bowled the ninth over and struck in second ball as Dhawan top-edged the googly to Shreyas Iyer, who covered good distance to take the catch.
Tight overs by Morris, Mishra and Yadav, created pressure and Williamson, trying to break free, went for a pull off Shami bouncer but only top-edged it to Morris.
There was gap of 15 balls when no boundary was struck. Yuvraj finally found one off Shami and it raised the team's 100 in 13.2 overs.
Runs were still tough to come by for SRH with Yuvraj and Henriques managing only singles and doubles. Yuvraj though changed the script after being dropped.