Faced with a 386-run first innings deficit, at tea on day three the West Indies had reached 107 for one with Brathwaite on 34 and Shimron Hetmyer on 27.
They still need a further 279 to make New Zealand bat again, with conditions much to the liking of the batsmen.
The wicket has browned off from the green cover on day one when the West Indies were rolled for 134 in their first innings.
Despite the Herculean task ahead to save the match, Brathwaite and his opening partner Kieran Powell were both prepared to take risks and attack the bowling.
Powell on 15 had an edge fall just short of Ross Taylor at first slip and a chip from Brathwaite on 19 fell short of Jeet Raval at short cover.
When the West Indies reached 41 after 15 overs, New Zealand's first innings destroyer Neil Wagner was brought into the attack with his short-pitched salvo.
But this time the tourists were primed to face him.
Powell smacked successive balls for six, and also took one to the helmet, in a costly spell which saw Wagner concede 25 runs in his first two overs.
Matt Henry, who bowled without reward on the first day, took the only wicket in the middle session when he had Powell caught and bowled for 40.
New Zealand had resumed the day on 447 for nine, with Blundell on 57, and as the West Indies struggled for inspiration, the nerveless debutant was ready to pounce.
He took 10 off the first over of the day from Miguel Cummins and progressed effortlessly from there.
Number 11 batsman Trent Boult assisted as he could, surviving two lbw appeals and a dropped catch as he staunchly preserved his wicket while Blundell closed in on the 100 mark.
When New Zealand eventually declared at 520-9, Blundell was not 107 off 180 deliveries while Boult had faced 60 for his 18.
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