The tourists are 67 for two chasing New Zealand's enormous first innings total of 609 for nine declared.
There are still three days' play remaining when the game resumes at the University Oval in Dunedin today and someone in the Windies batting line-up will have to follow Ross Taylor's lead.
The Black Caps batsman resumed on 103 and was undefeated on 217 when the captain, Brendon McCullum, declared shortly after the tea break.
Taylor batted for more than eight hours and resisted every instinct to play a slog sweep or have a dabble outside off stump. It is great watching him swat boundaries but yesterday's effort was a masterclass in discipline from a player who has relied heavily on his instincts in the past. He gave chances, of course. West Indies spearhead Tino Best almost claimed his wicket before Taylor got going on day one and he was dropped yesterday on 131.
But he batted on and on, just like the West Indies will need to do. The match got away on the visitors on Tuesday when they produced a poor bowling performance. Best was the exception to the rule. He ran in hard on both days.
''I think today we bowled a little better than yesterday,'' he said.
''Six hundred is not a reflection of how we bowled today but I think that yesterday was the problem. Three hundred and seventy runs on the first day in a test match is not good enough.''
Despite the signs looking bleak for the West Indies, Best said with some patience and judicious shot selection, there was no reason why the visitors could not occupy the crease for long periods and make New Zealand work for what would be a first test victory in a year.
''What we can take from the New Zealanders is they leave a lot of balls alone. Good balls I bowl at them they left alone ... We have to adopt that same attitude as well.''
Earlier, Taylor and McCullum combined in a New Zealand record fourth-wicket partnership against the West Indies of 195. The pair eclipsed the previous mark set by Nathan Astle and Matthew Sinclair in Wellington in 1999.
That partnership was broken early on when McCullum was dismissed for 113 - having added four to his overnight total. But Taylor found plenty of other accomplices. He teamed up with BJ Watling, Ish Sodhi and Neil Wagner in sizable partnerships as well to help guide New Zealand to its fourth-highest team total and highest when inserted.
Taylor is only the 13th batsman to score a double hundred for New Zealand.
Any thought of declaring during the tea break was dismissed, with Wagner saying the skipper wanted to take the West Indies further out of their comfort zone.
''Getting them to go back out and then giving their batters a minimal amount of time to pad up and make them feel a bit rushed is not always a good feeling. So that definitely worked a little bit,'' he said.
Left-armer Trent Boult struck in the third over, removing Kirk Edwards with a nice delivery which took the edge. Peter Fulton provided the finish at second slip.
Tim Southee made it 24 for two when Kieran Powell pushed out and nicked through to the keeper.
''We're definitely in a very good position. Two or three quick wickets [today] will definitely set us up,'' Wagner said.
''Everybody is fizzing for it. We really want to win a test and win the series. Everybody is working hard towards that.''
