Opener Georgia Plimmer led the chase with 53 and Amelia Kerr was brilliant with an unbeaten 34 and two wickets for New Zealand, who had to win in Sharjah to stay in contention.
Thery reached their target on 2-118 with 15 balls remaining after limiting Sri Lanka to 5-115.
Asian champions Sri Lanka crossed the 100-run mark for the first time in the tournament but rounded off their below-par campaign with four straight losses in group A.
Unbeaten group leaders and defending champion Australia have all but secured one semi-final berth.
But the other spot is still up for grabs. India play Australia on Sunday before New Zealand's last group match against contending Pakistan on Monday.Â
In an intense 38-degrees heat, Chamari Athapaththu led from the front with 35 off 41 balls with five fours, but once Kerr bowled the opposition captain in the 14th over, Sri Lanka struggled to hit boundaries in the death overs.
New Zealand struck through spinners Kerr (2-13) and Leigh Kasperek (2-27).
Plimmer's second T20 fifty came off 44 balls. With Suzie Bates (17), she shared a 49-run opening stand before she holed out in the outfield and gave Athapaththu a consolation wicket.
Kerr scored better than a run-a-ball 34 with three boundaries, and captain Sophie Devine knocked off the winning runs with the only six of the game.
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