Bangladesh originally set a target of 289 to win, which was achieved by the Black Caps in 49 overs.
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Martin Guptill was the star player in New Zealand's innings, hitting a ton from as many balls.
Ross Taylor added his half-century to the total, while putting on a 128 partnership with Guptill.
Runs began to become scarce as the Black Caps slowly approached their target.
The final few overs provided a tense finish but a massive six and a boundary from Daniel Vettori saw a grandstand finish, as New Zealand clinch their sixth victory from as many games.
The win puts them top of Pool A with 12 points, four in front of Sri Lanka and seven ahead of Bangladesh, who make up the top four.
Despite the loss, Bangladesh still progress to the quarter-finals alongside New Zealand, Sri Lanka and Australia in the group.