The visitors were bowled out for 293 after wasting a succession of starts by their middle-order batsmen and the needless run-out of all-rounder Jimmy Neesham when he looked to have seized back the momentum.
New Zealand won the first test in Jamaica before West Indies levelled the series.
Kiwi captain Brendon McCullum won his third successive toss of the series on Thursday and chose to bat but his opening pair failed again with Hamish Rutherford dismissed in the fourth over.
Kane Williamson (43), Ross Taylor (45) and McCullum (31) were unable to build on good starts and when BJ Watling (one) and Tim Southee (six) went in quick succession, New Zealand were struggling at 194 for seven.
Neesham, who scored a century in the first test, and off-spinner Mark Craig resurrected the innings with a 64-run partnership.
However, with the West Indies looking downcast and lethargic, the pair tried a sharp single when Neesham pushed the ball to short cover.
The all-rounder hesitated slightly, allowing Brathwaite the opportunity to run him out for 78.
"It was feeling good out there and I had a good partnership with Mark but I'm a little disappointed to fall short and (for us) to fall short of a par score as well," Neesham said.
"I think if we had got anywhere north of 320 or 330 it would have been a good effort, so to fall 20 or 30 short is a little disappointing.
"There's still a lot in it for the bowlers so if we get some early wickets tomorrow we're in it."
Craig, who scored 67 in the second test, was 46 not out when last man Trent Boult was stumped by wicketkeeper Denesh Ramdin for 12 to give left-arm spinner Sulieman Benn his fourth five-wicket haul in tests.
Benn finished with five for 93 while fast bowler Kemar Roach took four for 61.
(Reporting by Greg Stutchbury, Editing by Tony Jimenez and Frank Pingue)
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