Trent Boult captured the final West Indies wicket in the third over with the second new ball to give New Zealand a tense 53-run win in the deciding third Test in Barbados and a 2-1 series victory.
New Zealand won the first test in Jamaica, while West Indies won the second in Trinidad.
New Zealand's only previous series win in the West Indies was when they took a 1-0 victory in the two-match series in 2002.
The victory with 13.4 overs remaining in the final day's play also completed the third successive series win for Brendon McCullum's side after they beat West Indies 2-0 and India 1-0 at home in the southern hemisphere summer.
Boult had Jerome Taylor trapped in front for 12 to bowl the hosts out for 254, chasing 308 to win.
Facing the prospect of intermittent rain, McCullum had declared overnight at 331 for seven, a lead of 307, with Kane Williamson unbeaten on 161.
Windies' slow start
The West Indies were in trouble almost from the start of their second innings as medium-fast bowler Trent Boult dismissed Kraigg Brathwaite (six) and Kirk Edwards (10) cheaply.
Tim Southee subsequently sent the dangerous Chris Gayle packing for 11 with the last ball of the 10th over, which left the West Indies reeling at 31 for three.
Gayle's 33-ball resistance was ended whe he played the ball to his own stumpson via an inside edge and his pad.
A New Zealand victory or a draw seemed the only two likely outcomes from that stage though two short rain delays and then stubborn resistance from the tail threatened to save the home team.
Debutant fast bowler Jason Holder followed up his 38 in the first innings with 52 in a 77-run partnership with Shane Shillingford (30 not out) before offspinner Mark Craig bowled Holder between his legs off a delivery that kept low.
Last batsman Taylor, who scored a Test century against New Zealand in 2008, and Shillingford then put on 22 for the final wicket before Boult struck.