When the Black Caps were 208-3 in the 39th over they were on course for a total of 300, but the dismissal of the dangerous Corey Anderson triggered a collapse -- they lost three wickets for two runs in 11 balls -- and it needed Ross Taylor to hold the innings together.
The former captain finished 60 not out from 56 balls with just six fours, with wickets falling around him as New Zealand failed to build on a second-wicket stand of 133 between skipper Tom Latham and Neil Broom.
Latham, on the back of his century against Ireland in their big victory on Sunday, picked up where he left off and found the boundary 11 times in his 92 ball-innings of 84.
Broom also posted a half century before sweeping Nasir Hossain to square leg on 63. He faced 76 balls and hit seven fours.
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It was Shakib Al Hasan who started the collapse and Jimmy Neesham, Mitchell Santner and Colin Munro -- all big hitters -- followed in successive overs.
A Bangladesh win would promote them above Sri Lanka into sixth place in the ODI rankings, four months before the cut- off date which decides the top eight for the next ICC World Cup.