New Zealand's tail order showed its vulnerability with the last five batsmen making a combined 19 runs.
The start in their chase was not the best though, as opener Soumya Sarkar gave an easy catch in the third ball of the innings, but his fellow batsmen rose to the occasion to give the team the flawless boost before next month's Champions Trophy.
And it will back itself to pull it off, after an all-round effort with the ball and half-centuries from Tom Latham, the captain for this series, and Neesham took it to a four-wicket win with enough balls in hand in the first exchange against Bangladesh.
New Zealand have already sealed the tri-series title by winning all three of their previous matches, and while this game is academic as far as this ODI series is concerned, it is still pretty important in the context of the Champions Trophy. With September 30, 2017 being the cut-off date for direct qualification for the ICC Cricket World Cup 2019, Bangladesh are in a strong position to be one of the seven automatic qualifiers apart from host England to make them to the pinnacle event. Most recently, they drew an ODI series in Sri Lanka and finished second in the tri-series, involving Ireland and New Zealand.
Opting to bowl first, Bangladesh made an early breakthrough as Mustafizur Rahman dismissed opener Luke Ronchi for just two runs.
However, the duo returned to the pavillion in quick succession, and when Mosaddek Hossain and Shakib Al Hasan also fell cheaply, tension was running high.
Following Latham's dismissal, the Black Caps began to lose wickets in a hurry, but thanks to Ross Taylor's unbeaten knock of 60, which came off 56 balls and included six boundaries, New Zealand concluded their innings on 270/8.
Chasing 271, opener Tamim Iqbal (65) and one-drop batsman Sabbir Rahman (65) put Bangladesh on course before Mahmudullah (46*) and Mushfiqur Rahim (45*) helped the team register their first win against NZ overseas.
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