Shai Hope led the resistance on Thursday with his fourth Test century in an unbroken partnership of 140 with Justin Greaves which began when New Zealand captured four wickets relatively quickly in the second session.
At stumps, Hope was 116 not out, Greaves 55 not out and West Indies 4-212, trailing New Zealand by 319 runs in Christchurch.
The hosts' second innings came to an end at 8-466. It was a moot point whether New Zealand declared or simply ran out of fit batters. Wicketkeeper Tom Blundell was unable to front because of a hamstring injury and allrounder Nathan Smith was inactive with a side strain.
Matt Henry bowled 11 overs in West Indies' second dig before visiting the hospital, handily adjacent to Hagley Oval for scans on a painful calf muscle.
Jacob Duffy and Zak Foulkes, with only two previous Tests between them and in their first Tests at home, were left as the Black Caps' only fit pacers.
To add injury to injury, Hope is still affected by a serious eye infection which has forced him to bat in sunglasses in both innings; making a half-century in the first innings and a hundred in the second.
"I don't want to give away too much but it's just about understanding what you have to do to overcome a situation," Hope said.
"They've got quality bowlers in Test cricket, nobody's a walkover in this format."
Michael Bracewell (1-54) bowled 22 overs, Rachin Ravindra nine, Duffy 19 overs for figures of 2-65 and Foulkes bowled 13 overs.
New Zealand started day four already 481 ahead after the tourists were bowled out for 167 in reply to NZ's 231.
The Black Caps lost Will Young (23), Bracewell (26), Henry (8) and Duffy (10) before the innings ended with Foulkes 11 not out.
Kemar Roach dismissed Young, Bracewell and Henry to finish with 5-78, his 12th five-wicket bag in Tests. He is the oldest bowler at 37 to take five wickets in an innings against New Zealand and now has 291 Test scalps, placing him fifth on West Indies' all-time list.
After New Zealand's innings ended, John Campbell (15) and Tagenarine Chanderpaul (6) carried the West Indies through to lunch. Then Duffy removed both openers after the break. Bracewell removed Alick Athanaze (5) and Henry dismissed Roston Chase (4) leaving West Indies 4-72.
Hope and Greaves then batted through most of the last two sessions to see them to stumps and to set up an intriguing final day.
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