Jacob Duffy returned figures of 5-42 and Ajaz Patel 3-23 as the tourists folded chasing 462, while Brandon King top scored with 67 opening the innings.
Conway's 327 runs so far are the third highest for New Zealand in a single Test, behind Stephen Fleming's 343 (274 and 69) against Sri Lanka in 2003, and Martin Crowe's 329 (30 and 299) against Sri Lanka in 1991.
Having bowled out West Indies for 205, New Zealand posted 278 before being all out in the final session on day two at Basin Reserve.
"This is essentially like a knockout game. We have got to give it everything we can," Stead told New Zealand media on a video call ahead of a do-or-die clash against the West Indies.
