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.
Only the staunch defiance of Ross Taylor, who made 131 -- his third century in three consecutive Tests -- enabled New Zealand to reach 349 in reply to the West Indies' 367.
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.
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.
"There's genuine excitement," Taylor said of the prospect New Zealand will claim back-to-back victories.
"To keep (the target) down to 122 was outstanding. The way they (Boult and Tim Southee) bowled and the aggression they showed, they need a lot of pats on the back for the hostility they showed."
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.
Shivnarine Chanderpaul was back in the middle with the West Indies at three for 13 but he could not reproduce his century-making form of the first innings.
In another mini-breakthrough, Corey Anderson had Marlon Samuels (eight) caught behind and Chanderpaul (20) was caught by Kane Williamson diving to his right at third slip off the bowling of Neil Wagner.
After 18 overs, the West Indies were five for 46 and their buoyant mood at finding a pitch favourable to their spin attack had long evaporated.
Denesh Ramdin (18) and Darren Sammy (24) added some fight but the innings ended when Southee came back for a second spell and took the last three wickets in four balls.
Boult finished with four for 23 and Southee three for 12. New Zealand had Taylor to thank for being as close as they were at the end of the first innings.
He batted for nearly five-and-a-half hours for his 131 which featured 16 fours and two sixes. For most of his time in the middle Taylor was a model of precision and patience, deftly picking up ones and twos.
He did show aggression towards the quicks when the second new ball was taken, hitting Tino Best for four to bring up his hundred and clouting 18 off a Darren Sammy over, before being removed when he cut a wide Best delivery to Marlon Samuels.
Taylor has scored 864 runs from 16 Test innings so far this year at an outstanding average of 72.00. In this series he has scored 493 in four innings and only been dismissed twice.
BJ Watling chimed in with 20 before he was caught behind off Sammy, leaving Narine to remove Tim Southee (18), Ish Sodhi (nine) and Neil Wagner (22) to wrap up the innings.
Despite the gloomy outlook for the tourists, Narine banked on an even better performance in New Zealand's second innings.
"Wickets send jitters, hopefully, you never know, we can get a couple of wickets and go from there," he said.
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