After starting their second innings with a 386-run deficit, by stumps it had been reduced to 172 as the Calypso cricketers clocked off for the day at 214 for two.
Opener Brathwaite, with a seventh century in his sights, was on 79 with Shai Hope on 21.
The Basin Reserve wicket has flattened out and lost the greenness that saw 19 wickets fall in the first two days.
The only wickets on day three were the West Indies' Keith Powell (40) and Shimron Hetmyer (66) after New Zealand resumed the day at 447 for nine and declared at 520, still with a wicket in hand, and with debutant Tom Blundell unbeaten on 107.
"He is one of the most important batsmen in the team as of now. It feels better, it gives us drive to know that there is Kraigg there so you can be free at your end to play shots," 20-year-old Hetmyer said.
"I am very confident batting with Kraigg because he keeps telling me just be yourself and express yourself and enjoy it as much as you can.
Matt Henry took the two wickets for an economical 33 runs while Neil Wagner, the first innings destroyer, kept the batsmen ducking and weaving but was expensive at none for 89.
While Brathwaite offered the West Indies hope, Blundell felt he was an accidental hero who began the day unbeaten on 57 and with no inclination he would end up a century-maker.
"Not at all to be honest, I was just going to go out there and let it play out, then I looked up at the scoreboard and I was 85 and then I was like I've got a chance here," he said while acknowledging the role of batting partner and number 11 batsman Trent Boult.
While the West Indies faced a monumental task to save the Test, Brathwaite has the memory of the West Indies' last tour of New Zealand to know it is not impossible.
In the first Test on the 2013 tour they were 396 in arrears when forced to follow on.
A Marlon Samuels double century took them to 507 in their second innings on the fifth day and New Zealand ran out of time chasing a target of 112 to win.
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