Latham made the call at drinks in the third session on Sunday with the hosts leading by 461 runs. The Black Caps earlier had bowled out West Indies for 420 in reply to their 8-575, a first innings lead of 155.
At stumps after 16 overs the tourists were 0-43 with Brandon King 37 not out and John Campbell unbeaten on two. They will start the final day needing 419 to win on a pitch on which cracks and creates inconsistent bounce.
Latham and Conway shared a 323-run opening stand in New Zealand's first innings and became the first pair in Tests to achieve a triple-century and century partnership in the same match.
Conway followed his first innings of 227 with 100 to become only the 10th player and first New Zealander to score a double-century and ton in the same Test.
Latham made 137 in the first dig and 101 in the second, emulating his father Rod who joined with Mark Greatbatch to make century opening partnerships in both innings of a 1992 Test against Zimbabwe.
In this series Conway scored 452 runs at 75.3, while Latham totalled 427 at 71.1. Their partnerships in this Test produced 515 runs, more than any opening pair in a single match in history.
"I think all the work the guys have been doing for a while now is paying off the way they want," New Zealand batting coach Luke Ronchi said.
"I think the mindset is where they want it to be and the trust in their own game and the game plans against particular bowlers is coming off.
"But it also a nice batting wicket as well, not the traditional sort of New Zealand wicket you see in most test matches."
Earlier in the day, Jacob Duffy took two quick wickets to help bowl out West Indies in the first hour of the fourth day.
Duffy took 4-86 in the first innings to go with his 5-34 and 3-122 in the first Test and 5-38 and 4-86 in the second.
