Four batsmen posted half centuries as New Zealand overhauled the West Indies' total of 301-8 to win the deciding one-day international by five wickets.
The win was an historic one for the Black Caps, whose victory in Barbados on Monday lifted them to a 2-1 series win and makes this their first ODI series win in the Caribbean.
Tom Latham scored 69 and put on 120 for the fourth wicket with Daryl Mitchell (63) in a stand which paced the run chase and carried New Zealand in sight of victory.
Martin Guptill (57) and Devon Conway (56) had earlier combined to add 82 runs for the second wicket to give New Zealand a platform for a demanding chase.
Mitchell was out with the total 248 in the 42nd over and Latham at 259 in the 44th, still with New Zealand requiring 43 runs. Jimmy Neesham settled the issue with a decisive innings of 34 from 11 deliveries, taking 18 runs from the 45th over bowled by Yannic Cariah.