Hamish Bennett, the one change in the Black Caps line-up which beat Ireland in their first match on Sunday - he replaced Scott Kuggeleijn - was rewarded for an economical 10-over spell with two wickets in the last over to finish with three for 30.
Winning the toss, Black Caps skipper Tom Latham asked the Tigers to bat first and openers Tamim Iqbal and Soumya gave Bangladesh the flawless start.
Latham's pleased with the way they timed their chase, and said they managed to build a lot of partnerships and that was the pleasing thing.
With the opening game between Bangladesh and Ireland rained out, the Black Caps can get two wins clear with a victory tonight.
Having been asked to bat first, three Bangladesh batsmen posted half-centuries, with Soumya Sarkar hitting 61, Mushfiqur Rahim 55 and Mahmudullah 51.More news: Will give my best in Champions Trophy, says Mohammed Shami
New Zealand skipper Tom Latham scored a solid 54 runs off 64 balls with Jimmy Neesham chiming in with 52 runs off 48 balls.
JDS Neesham (New Zealand) declared man of the match.
Bangladesh are likely to retain much of the side that lost Wednesday's match against New Zealand by 15 runs.
Mashrafe Mortaza (1-58), who returned from a suspension due to poor over-rate opened the bowling for Bangladesh but he was given a raw deal by Luke Ronchi (27), who hit a six and a four in the first over.
Just when Bangladesh were looking good crossing the 115-run mark in the 25th over, Sarkar was caught at mid-wicket to give Ish Sodhi (2-40) a wicket.