West Indies need 462 to win after Conway's historic century
Conway's 100 followed his 227 in the first innings, and made him just the sixth New Zealander to score a hundred in both innings of a Test.
Moments later Tom Latham became the seventh, moving from 80 at tea to 101, having scored 137 in the first innings.
Latham and Conway had a mammoth 323 opening partnership in the first innings, and set a world record for the opening wicket in combining for 515 runs in the match.
Conway joins an illustrious list in scoring a double century and a century in the same Test match.
Only Doug Walters, Greg Chappell, Marnus Labuschagne, Sunil Gavaskar, Shubman Gill, Lawrence Rowe, Brian Lara, Graham Gooch and Kumar Sangakkara had achieved the feat.
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.
Conway and Latham added 157 runs in the middle session at more than five an over as they looked for quick runs to set up a declaration before the close.
Kane Williamson made 40 from 37 balls and Rachin Ravindra hit four sixes as he blasted 46 from 23 balls, ticking New Zealand over the 300 mark.
West Indian century maker Kavem Hodge continued a fine personal performance, taking 2-80 from 17 overs with his left-arm spin.
Captain Roston Chase was hammered for 106 runs in just 13 overs.
New Zealand took four wickets in the morning session as West Indies were 420 all out, with Hodge left stranded unbeaten on a fine 123 from 275 balls, including 15 boundaries.
Jacob Duffy finished with 4-86 including the first two wickets of the day, taking his tally for the three-match series to 18.
The West Indies resumed on 381-6, with Hodge on 109 batting alongside Anderson Phillip.
Phillip lasted just nine balls and Shai Hope, who missed the last two days because of food poisoning, fell for four.
Jayden Seales briefly showed some resistance making 15 from 53 balls until he was bowled by Ajaz Patel with a ball that grazed the off stump on the way through to the keeper.
Kemar Roach was the last man to bat, nursing a hamstring strain suffered on day one, but lasted only two balls.
Patel took 3-113 in 41 overs, his first wickets on home soil, while Michael Rae had 2-89.
New Zealand made 575-8 declared in their first innings.