Sri Lanka had three wickets in hand and required 13 off the last over by Grant Elliott at Mount Maunganui but could only manage 10 runs as they lost two wickets.
The result put the pressure on Sri Lanka to win the second and final game in Auckland on Sunday or they will slip from first to third behind the West Indies and Australia in the world rankings.
New Zealand, sent into bat first at Bay Oval, made 182 for 4 in their 20 overs and Sri Lanka replied with 179 for 9.
Tillakaratne Dilshan went at the start of the second over and captain Dinesh Chandimal was out in the third over as they tried to force the pace against the bowling of Matt Henry and Trent Boult.
Milinda Siriwardana (42) and Thisara Perera (28) started to repair the innings with a stand of 49 in 27 balls for the sixth wicket and Chamara Kapugedera (17) and Nuwan Kulasekara (14) kept up the pressure.
But on the third ball of the final over Kulasekara was well caught in the deep by Boult and Kapugedera was run out on the next delivery.
But they had no sooner passed 100 when Guptill was first to fall for 58 when he was run out by a side-on throw from Kapugedera on Williamson's call for a quick single.
Williamson followed soon after for 53 off 42 while Colin Munro scored 36 and Ross Taylor was not out 22.
For Sri Lanka, Kulasekara took two for 26, while the leading bowlers for New Zealand were Boult with three for 21 and Henry with three for 44.
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