Chasing 292, after skipper Kane Williamson's fine 148 for New Zealand, West Indies were well poised at 130 for two after 20 overs but never recovered from the loss of Shimron Hetmyer (54) and Chris Gayle, who made 87 including six sixes.
West Indies collapsed to 164 for seven and looked out of the game until Brathwaite led a rearguard action, bringing back memories of his sensational performance in the 2016 World Twenty20 final.
Brathwaite's power-hitting, including nine fours and five sixes and a 41-run 24-ball last-wicket partnership with Oshane Thomas who did not score, put West Indies within five runs of victory.
But on the final ball of the penultimate over, Brathwaite went for the big hit, only for his drive off Jimmy Neesham to find Trent Boult's hands at deep mid-on and the West Indies batsman collapsed to his knees.
The win moved unbeaten New Zealand to the top of the table with 11 points and left West Indies staring at likely elimination in seventh place with three points.
(Reporting by Simon Evans, editing by Ed Osmond)
