Marlon Samuels, who scored a crucial 123 in the first innings, went about steadying the West Indies' run chase as he and Assad Fudadin put on 74 for the third wicket.
Fudadin was dismissed late in the day when he played on a Neil Wagner delivery and when Doug Bracewell had Samuels caught in the slips, the Kiwis found themselves back in the match.
Much will rest on the shoulders of West Indian veteran Shivnarine Chanderpaul when day four resumes, with the left-hander unbeaten on 20.
Earlier in the day, New Zealand began the morning on 59-2 but failed to take advantage of their 51-run first innings lead as they stumbled to 154-all out in their second dig.
Only Dean Brownlie offered any major resistance as he chipped away to score 35 but none of his partners managed to pass 20 and New Zealand blew a golden chance to build a decent lead and set the West Indies a huge chase for victory.
It was the same case of the batsmen failing when they needed to step up, with Brendon McCullum (19), Ross Taylor (0) and Kane Williamson (8) all missing out.
West Indian spinners Narsingh Deonarine (4-37) and Sunil Narine (3-19) did the damage as they combined to take seven wickets.
Narine was particularly miserly as his 12 overs only conceded 19 runs as he was able to build pressure on the New Zealanders.