The West Indies' first innings folded soon after lunch on the third day in Dunedin and they were asked to bat again, trailing the Black Caps by 396 runs.
By tea they had reached 37-1 in their second innings, with Kirk Edwards unbeaten on 12 and Darren Bravo on 10.
New Zealand quicks Tim Southee and Trent Boult did the bulk of the damage in the first innings, taking seven wickets between them.
Only Shivnarine Chanderpaul, playing in his 151st Test, offered any real resistance, reaching 76 before he was trapped leg before wicket by Boult.
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It was Chanderpaul's 62nd Test half-century and he also passed the 11,000-run milestone during his innings.
The West Indies started the day on 67-2. Bravo (40), Marlon Samuels (14) and Narsingh Deonarine (15) fell to Southee while Boult claimed the scalps of
Denesh Ramdin (12) and Chanderpaul.
Southee finished the innings with figures of four for 52 and Boult took 3-40.