Shakib Al Hasan remained the top-scorer with 114 of just 115 balls while he was accompanied with Mahmudullah who scored an unbeaten 102 of 108 balls including eight fours and two sixes. "Shakib took some time and then he took off", Mahmudullah added.
Chasing a modest target of 266, the Tigers were restricted to 33 for four after Mushfiqur Rahim got out in the 12th over.
To make it to their first ever semi-final in an ICC tournament, Bangladesh will hope that England beat Australia on Saturday, which would leave Australia on 2 points and Bangladesh will advance three points from 3 games.
For Bangladesh, Mosaddek scalped three wickets while Taskin Ahmed took two wickets each.
Ross Taylor top scored for the Oceanian team with 63 while captain Kane Williamson hit 57 runs.
The Kiwis barely saw out their 50 overs, as they lost six wickets for only 64 runs in nearly 12 overs at the end after winning the toss and choosing to bat first. Tournament hosts England have already made it to the semis by beating New Zealand and Bangladesh.
The two sides have just a point apiece in the eight-team marquee tournament after they slumped to a comprehensive defeat by England, while their respective matches against two-time victor Australia were abandoned.
Guptill was the next batsman to go in the 13th over as he was beaten by a Rubel Hossain delivery to be out LBW.
The winning moment came when Mosaddek Hossain, Bangladesh's hero with the ball, flashed a boundary through third man off Milne.
They bowled tightly and weathered the early attack from opener Martin Guptill, who struck New Zealand's only six in the fifth over of the game.
The live broadcast of the New Zealand Vs Bangladesh Champions Trophy match will start at 9.30 pm GMT (3 pm IST).
The toss was delayed an hour after heavy overnight rain left the run-ups at both ends too soggy, but had dried out enough for play to start 60 minutes late, but with no overs lost.
Lower middle-order batsmen Corey Anderson (0), Mitchell Santner (14 not out), Adam Milne (7) and Tim Southee (10 not out) then failed to step up to the occasion as they failed to reach a handsome score.
"We knew it was going to be hard because conditions weren't that easy".
There was a brief lull after the equation was down to 70 off the last 10 overs, but Shakib and Mahmudullah opened up again with cuts, chips, full-blooded drives and an upper-cut to bring the asking rate down to below a run a ball.
Both Bangladesh and New Zealand are in the last-chance saloon in Cardiff on Friday, with the loser definitely heading home.
"I think once they get out in the middle, things will just click into place, and yeah, hopefully we can play well in Birmingham", added Smith of a line-up where the big-hitting Chris Lynn, fresh from his IPL exploits, could be recalled.