The left-arm seamer became the second Englishman, after Chris Jordan, to claim a hat-trick in T20 Internationals as England bundled out Sri Lanka for 133 with four deliveries still left in the innings.
Phil Salt led England's successful chase with 46 off 35 balls as the tourists clinched the low-scoring match that had been reduced to a 17-overs-a-side contest following rain.
Earlier put into bat, Sri Lanka cruised to 76-1 in seven overs with Kusal Mendis (37) and Pathum Nissanka (23) in the middle.
Spinner Adil Rashid (3-19) triggered a batting collapse as he removed both the set batters and Dhananjaya de Silva too as Sri Lanka slumped to 98-6 in the 12th over.
Curran mowed down the Sri Lankan tail in the eventful 16th over with his hat-trick.
Dasun Shanaka was snared at mid-off, Maheesh Theekshana perished at long on, and Matheesha Pathirana had his stumps rearranged. Sri Lanka lost their last four wickets for four runs in a spectacular meltdown.
Jofra Archer, returning from a side strain, returned figures of 3-0-22-0.
When England came out to bat, Jos Buttler (17) hit Ehsan Malinga for four fours in a row before losing his stumps to the pacer.
Malinga also removed Jacob Bethell, and Salt could have been dismissed too, but Kamil Mishara dropped a sitter at deep mid-wicket.
Tom Banton fell after a breezy 29, but Sri Lanka's sloppy catching was on display again as Nissanka dropped a regulation catch in the deep in a major relief to England captain Harry Brook when the batter was on 11.
Shanaka eventually dismissed Salt in the 15th over, but England were on the verge of victory by then.
England were 125-4 in 15 overs when rain stopped play with the visitors comfortably ahead of their target at that stage.
The second T20 International of the three-match series is scheduled on Sunday.
