The tourists ended day three at Port of Spain on 1-73 in their second innings, still 166 runs behind.
The Black Caps dismissed their hosts for 460 just before tea to trail by 239 runs.
They then lost skipper Brendon McCullum in the second over of the final session.
McCullum had promoted himself from the middle order after opener Hamish Rutherford failed to take the field earlier in the day because of a stomach complaint.
But having got to three, he was rapped on the pads and given out leg-before.
McCullum called for a review and the replay went firmly against him, showing the ball would have struck the inside of the leg stump.
Tom Latham and Kane Williamson made slow but steady progress after that and by stumps had put on an unbeaten partnership of 64.
Latham continued his good form by ending the day on 30, and Williamson was not out 38.