Captain Shai Hope top-scored with 53 from 39 balls as the West Indies made 6-164 after losing the toss and batting first on a pitch that was two-paced and unpredictable.
New Zealand were going well at 2-70 in the 10th over before losing 7-37 to slump to 9-107 in the 17th over.
The match seemed over but New Zealand captain Mitch Santner fanned home fans' hopes with an outstanding display of power hitting as he made 55 from 28 balls.
Santner hit four fours and a six from the 18th over bowled by Matthew Forde, then three fours from the first three balls of the 19th over from Jason Holder.
New Zealand came to the final over needing 20 to win and Santner hit the third ball from Romario Shepherd for six.Â
But Santner couldn't keep up the assault, taking only two runs from the next two balls before hitting a boundary from the final delivery.
"I think the way the West Indies bowled in the middle there, they really put pressure on us," Santner said.
"I guess we weren't really able to go anywhere. We were reasonably happy with 164 at the turn, but it was one of those nights I guess."
The West Indies came to New Zealand from their series in Bangladesh and it seemed likely they would struggle in very different conditions where the ball swings and the bounce is more pronounced.
Eden Park usually provides pace and bounce, but Wednesday's pitch was originally prepared for a rained-out Twenty20 against England last month and lacked its usual zip.
It produced a major challenge for all batters when the ball was banged into the surface.
The West Indies lost Brandon King (3) to the last ball of the first over of their innings from Jake Duffy.
Alick Athanaze (16), who was lucky to avoid being run out at the non-striker's end when he was on two, was eventually caught by Mark Chapman off Kyle Jamieson when he skied a pull shot.
The West Indies were 2-32 after the first power play.
Hope reached his ninth T20 half century with a six off Zak Foulkes's bowling, then was clean bowled next ball swinging across the line at a delivery that stayed low.
Roston Chase (28) and Rovman Powell (33) put on 49 for the fifth wicket. Powell hit two fours and two sixes in a 23-ball knock, giving impetus to the innings as the West Indies added 53 in the last five overs.
New Zealand were 1-48 after the power play, losing Devon Conway (13) in the fourth over to a ball from Forde that was angled sharply in to the left-hander and pierced his tentative defensive shot. Tim Robinson (27) was out to the first ball after the power play.
New Zealand then became bogged down. Boundaries were rare and they lost wickets regularly as the Black Caps fell behind the run rate.
Chase bowled four excellent overs of off-spin to claim 3-26 and Jayden Seales took 3-32.
The teams meet again at the same venue on Thursday.
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