The hosts had slipped to 20 for three in the seventh over but their middle-order batsmen all made useful contributions to help the team reach 264 before they were dismissed off the penultimate ball of their 50 overs.
Kieron Pollard top-scored with 56 but Marlon Samuels (46), Devon Thomas (37), Darren Sammy (26) and Andre Russell (29) all helped ensure the West Indies had a reasonable total to defend.
The West Indies suffered a heavy 88-run defeat in Wednesday's third ODI to see their lead in the five-match series trimmed to 2-1 and they would have been eager to make a good start after winning the toss and choosing to bat.
However, they soon found themselves in trouble as Johnson Charles (one), Chris Gayle (16) and Dwayne Smith (nought) all fell in the space of just three runs.
The home side then found themselves 59 for four when Dwayne Bravo fell for 18, but Pollard, Samuels and Thomas helped repair the damage before Sammy and Russell, who between them struck four sixes, provided some late momentum.
Jacob Oram (three for 42) and Tim Southee (three for 53) were the most successful Black Caps bowlers.