Chasing a modest 136 to win, Sri Lankaâs batsman struggled to capitalise against the second-string visiting team.
But a crucial 38-run stand for the sixth wicket between Asalanka and Wanindu Hasaranga steadied the ship and inched the team to victory.
âOverall it is a good win,â Asalanka said.
âWe should have restricted them to 110 or 120 but they did well with the lower order.â
The stand was broken when Hasaranga went for a big shot off Zakary Foulkes but fell short, caught at mid-on by Kiwi captain Mitchell Santner and departing for 22.
Left-arm spinner Dunith Wellalage, who took three of New Zealandâs wickets after the tourists opted to bat, sealed Sri Lankaâs win in style when he pulled Jacob Duffy for a six over mid-wicket.
Asalanka finished unbeaten on a well-judged 35 off 28 balls.
New Zealandâs below-par total had given Sri Lanka some breathing room.
Top-order batsman Kusal Perera etched his name in the record books, surpassing Tillakaratne Dilshan to become Sri Lankaâs highest run-scorer in T20s with his quickfire 23 off 17 balls.
New Zealandâs 135 all out was the third-lowest total in T20Is in Dambulla.
âIt is a challenge to adjust to conditions,â Santner said.
âAt halfway we felt that we can defend it but a couple of partnerships took it away from us.â
âWe were expecting 160 after winning the toss but the dew settled in and it was hard to hold the ball and thatâs probably food for thought tomorrow.â
Fresh from a commanding red-ball series win in India, the tourists arrived in the island nation missing Tom Blundell, Rachin Ravindra and other stalwarts who are resting for an upcoming Test battle against England.
All-rounder Michael Bracewell showed some fight with 27 off 24, including two fours and a six.
New Zealand managed to go past 100 after a gutsy 39-run stand for the ninth wicket between Ish Sodhi and Foulkes.
Foulkes, joint top-scorer with 27 off 16, hit a four and a six to help the Kiwis salvage some pride before Sodhiâs departure on 10 in the final over.
New Zealand handed a debut to wicketkeeper-batsman Mitchell Hay, who was caught off a Matheesha Pathirana delivery on his second ball for nought.
The second and final match of the series is at the same venue on Sunday.
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