Sri Lanka got off to a fine start, with Niroshan Dickwella immediately finding his touch.
He reeled off seven boundaries in his first 12 deliveries, scoring all around the ground as the hosts raced to 38 without loss.
Chris Woakes, who has not played a T20 for England since 2015, was retained for a game that was one over longer and saw his opening two-over burst leak 21.
Dickwella offered a half chance when sweeping Curran on 28 but Adil Rashid could not get a hand on it at short fine-leg.
Sri Lanka lost their first wicket with 57 on the board in the sixth over.
Curran was the bowler, drawing a toe-end mis-hit from the dangerous Dickwella that sailed high to mid-off. Woakes had a tricky job gathering a slippery ball as it dropped over his shoulder but he held on at the second attempt.
Adil Rashid came on to bowl the seventh over and quickly nabbed England's second success, slipping a brilliant googly between Mendis' bat and pad and cleaning the recalled batsman up for a golden duck.
After 11 overs - just over halfway - Sri Lanka had moved to 87 for two.
They had been unable to rediscover Dickwella's early bravado but both Sadeera and Dinesh Chandimal were finding way to keep the scoreboard moving.
The game turned England's way in the 13th over, with Rashid taking two wickets with successive balls.
His first was hardly a collector's item, Sadeera giving his hard-fought innings of 35 away to a rank full toss which he steered inexplicably to square leg.
Thisara Perera's first-ball dismissal was more conventional, though not much smarter. He looked to heave a conventional leg-spinner for six down the ground, failed to get enough of it and was impressively caught by a sprawling Jason Roy.
Sri Lanka were 95 for four at the end of of the over and losing momentum fast.
With five overs of the innings left Sri Lanka were 110 for four, with Ben Stokes having turned in a superb selection of slower balls that baffled Dhananjaya de Silva.
The England all-rounder's first three overs had cost a miserly 10 runs as he showed a selection of variations.