After four days of bright sunshine, the heavy downpour began at tea on the final day when New Zealand were 79-4 chasing 112 for victory.
After wrapping up the West Indies second innings for 507 before lunch, the modest target appeared a formality despite the forecast rain.
However, the New Zealand batsmen faltered from the start and were rapidly reduced to 44-4 before Ross Taylor and Corey Anderson arrested the slide.
"We would have won that, I believe, if it hadn't rained," said a dejected Brendon McCullum who is yet to taste success in 10 Tests since replacing Taylor as captain after their last win in Sri Lanka in November last year.
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"It was disappointing to set ourselves up and be 33 short."
West Indies skipper Darren Sammy described it as a "moral victory" for his side coming off two innings defeats in India.
The New Zealand top order failed to read spinner Shane Shillingford, who was inconsequential in the first innings but produced exceptional figures of four for 26 off 15 overs on a well-worn pitch in the second.
He struck first in the last over before lunch when wicketkeeper Denesh Ramdin snapped up a sharp catch down the leg side to remove Peter Fulton.