After rolling New Zealand for a modest 156 in 42.1 overs, the tourists made hard work of reaching their target for the loss of eight wickets in 27.3 overs.
A rollicking unbeaten 43 from Darren Sammy in 27 balls took them over the line as wickets fell regularly at the other end.
When just the last two wickets were left, Sammy knocked off the final 10 runs with a six and a four off Mitchell McClenaghan to be sure of the win.
The victory, no matter how narrow, was a change of fortunes for a side comprehensively outplayed during their 0-2 Test series loss to New Zealand.
"There is some work to do with our batting but a win is a win and it will ease a lot of pressure on us," captain Dwayne Bravo said.
"Darren Sammy is leader of the fielding group and I think the energy in the field was what set the scene for the win for us, and he finished it with the bat."
Despite the narrow margin at the end, New Zealand captain Brendon McCullum said his side did not deserve to win.
"We kept picking up wickets at key times and in the end it was a pretty close game but it was a thoroughly deserved win for the West Indies. They outplayed us in all three facets of the game," he said.
Although the West Indies' batting remained fragile, the introduction of one-day specialists unleashed a bowling line-up that New Zealand failed to handle on a seam-friendly wicket.
Any confidence New Zealand took from the Tests quickly evaporated when they were sent into bat and faced new arrivals Ravi Rampaul, Dwayne Bravo and Jason Holder.
It was not the return to international cricket that big-hitting opener Jesse Ryder wanted.
He only faced five balls before slashing at a wide Rampaul delivery to be caught by Darren Bravo at cover point without scoring.
His failure signalled the start of a disappointing batting performance only redeemed by the McCullum brothers Brendon (51) and Nathan (47).
They contributed 98 of the New Zealand total, with the remaining nine batsmen scoring a total of just 49 runs with nine extras.
Like New Zealand, the West Indies top order could not establish a partnership as Johnson Charles (nine), Darren Bravo (14) and Dwayne Bravo (12) all fell to McClenaghan, who finished with figures of 5-58.