Brathwaite reached his hundred, from 80 balls, in the next over, leaving West Indies needing six to win.
In their last match against South Africa, skipper Kane Williamson played a match-winning knock of 106 not out to guide New Zealand home in a tense final over finish while chasing a modest 242-run target.
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 cameo contained five sixes and nine boundaries and took the match to the wire, sinking to his knees when he saw Boult hold the catch in the deep.
With six needed for victory, Brathwaite attempted to clear long on but found Trent Boult on the ropes instead, crushing West Indies' hopes of a much needed win.
New Zealand sit near the top of the table half way into the group stages and with good reason.
West Indies (WI) Playing 11 (Probable): Chris Gayle, Shai Hope (WK), Evin Lewis, Nicholas Pooran, Shimron Hetmyer, Dareen Bravo, Jason Holder (C), Ashley Nurse, Shannon Gabriel, Sheldon Cottrell and Oshane Thomas.
A New Zealand win seemed a long way off when Sheldon Cottrell trapped Martin Guptill lbw with the first ball of the match and followed it up by bowling Colin Munro with a flawless yorker later in the over.
Left-arm seamer Boult was the best bowler with four for 30 while Ferguson finished with three for 59.
Jason Holder nicked Ferguson's next ball and when Gayle followed as the third dismissal in a dozen deliveries, needlessly picking out a boundary fielder trying to smite a seventh six, 24 hours after Moeen Ali had done the same to precipitate England's demise to Sri Lanka at Headingley, hope was frittering away.
But then came a dramatic rally, with the game turned on its head by a stunning 48th over from Henry that yielded 25 runs.
It fell to Jimmy Neesham, also expensive, to bowl the next over.
Brathwaite then hit Neesham for two to complete a century. He was not done yet. In the end, the Black Caps managed to win the encounter by a slender margin of five runs.
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NY undercut the president's claim about never meeting Carroll with a photograph depicting him talking to her at a party in NY . Also in the photo were Trump's then-wife, Ivana Trump and Carroll's then-husband, NY newscaster John Johnson.
Mohammed Shami strikes with a hattrick
He, however, said such tough games against strong teams like India will do a world of good for Afghanistan . India , known for preserving wickets and accelerating later, managed only 41 runs in the first 10 overs.