At stumps at the end of the third day, New Zealand were six without loss in reply, needing 122 to win.
The West Indies, already down 1-0 in the series, were all out in their second innings for 103, after taking an 18-run lead on the first innings.
The dramatic collapse came a day after the West Indies claimed control of the Test when Sunil Narine's spin variations mesmerised the New Zealanders.
Narine finished with career-best figures of six for 91, but the gloss was quickly taken off his achievement by yet another batting collapse as New Zealand traded spin for pace and swing.
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Trent Boult, the chief destroyer when New Zealand won the second Test, removed the West Indies top order with three wickets for one run in the space of 13 deliveries.
Kraigg Brathwaite (seven) fell to a ball that shot back between bat and pad, Kieran Powell (nought) edged a ball to third slip and Kirk Edwards (one) was caught down the leg side.