Best's four seals series for Windies
New Zealand skipper Ross Taylor smashed a fine century but it was not enough to prevent West Indies wrapping up the one-day series with a 24-run win in Saturday's fourth match at Warner Park.
Taylor, playing in his first match of the series after injuring his shoulder in an earlier Twenty20 international between the two sides in Florida last month, made 110 off 115 balls.
It was an innings which included six boundaries and five sixes for his sixth one-day century.
But as his team chased a 267 target, he was outshone by West Indies seamer Tino Best who marked his recall to the team with 4-46, a performance which included the wickets of Taylor and the dangerous Jacob Oram.
Earlier, Kieron Pollard hit a patient half-century as West Indies recovered from 105-5 to make 264 all out.
Pollard top-scored with 56, Marlon Samuels made 46 while the recalled Devon Thomas hit a useful 37.
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Victory gave West Indies an unassailable 3-1 lead in the series which concludes with the fifth and final match at Warner Park on Monday.
"It's a good feeling indeed," said West Indies skipper Darren Sammy.
"The good thing for us today was that the lower order came through. New Zealand improved as the series went on, but it was good to see we kept our cool."
West Indies off-spinner Sunil Narine, who took 2-20, was named man of the match and he won rich praise from Taylor.
"You have got to give credit to West Indies," said Taylor. "They were under pressure. We would have liked to put the foot down on the throat, but we didn't execute well.
"You have got to back yourself to get 79 off the last 10, but you have to give credit to Narine."
West Indies were helped by an untidy performance in the field by New Zealand whose bowlers sent down 20 wides.
The home side lost Johnson Charles (1), Chris Gayle (16) and Dwayne Smith (0) by the seventh over with their team on 20.
After Dwayne Bravo had gone for just 18, Samuels and Pollard added 46 for the fifth wicket before Pollard and Samuels put the West Indies back on the road to recovery with a partnership of 85.
Thomas hit three fours and a six in his 43-ball stay while Pollard smashed five boundaries and a six off 70 deliveries.
Sammy (26) and Andre Russell (29 with two big sixes) added valuable late runs.
WEST INDIES: J Charles, CH Gayle, DR Smith, MN Samuels, KA Pollard, DJ Bravo, DC Thomas (wk), DJG Sammy (capt), SP Narine, TL Best, AD Russell
NEW ZEALAND: RJ Nicol, MJ Guptill, KS Williamson, LRPL Taylor (capt), TWM Latham (wk), JDP Oram, BB McCullum, NL McCullum, TG Southee, DAJ Bracewell, TA Boult