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.
Bangladesh collapsed to 12 for three and were facing a humiliating loss at 33 for four in the 12th over.
265 for eight still represented a stiff ask for the Tigers, who struggled against the New Zealand pace attack on a wicket that offered plenty of assistance, mainly by way of quick and steep bounce and carry, and Tim Southee (3-45) and Adam Milne (1-58) ripped through the top four.
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.
Electing to bat, New Zealand were nicely set up for a challenging total by captain Kane Williamson (57 off 69 balls) and Ross Taylor (63 off 82 balls) with solid half centuries but their good work were undone later as wickets tumbled in a heap in the slog overs.
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"Mosaddek [Hossain] came in and took three important wickets and Taskin [Ahmed] also bowled well", said Mahmudullah, who scored his first century since his back-to-back tons in the 2015 World Cup.
Boult should have got in on the act when Ross Taylor, injuring himself in the process, spilled Mushfiqur's hard edge to slip.
Bashar was also optimistic about England's victory over arch-rival Australia which will earn his side a spot in the semi-final. Both the teams went into the match without even knowing whether a win will guarantee them a place in the semi-finals.
Bangladesh bowled really well in the last ten overs when the Black Caps could score only 62 runs despite being well placed at 201 for four in 39 overs.
An incredible 200-plus stand between Shakib Al Hasan and Mahmudullah helped Bangladesh over the line and gave them a fighting chance at reaching the knockouts.
"When it's not seaming and it's not swinging, obviously we need to look at different measures", New Zealand paceman Trent Boult told reporters at Cardiff on Thursday.
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These include cutting all links with Iran , the pan-Arab Muslim Brotherhood network and the Palestinian militant group Hamas. "No one has the right to intervene in our foreign policy", Sheikh Mohammed bin Abdulrahman Al-Thani said.
"It wasn't just the fact that Bangladesh defeated New Zealand by five wickets, but the manner and the circumstances under which victory was achieved, that made it truly special".
Earlier, a wet outfield due to heavy overnight rain and showers scudding across the ground this morning delayed the start of the play.
Earlier New Zealand batted first after winning the toss and posted 265 runs on the board.
Mahmudullah joined Shakib at the crease and the pair simply picked off any loose deliveries and continued to rotate the strike.
"It was frustrating to be on the other side of it, we wanted to come into this game firing but we didn't play our best cricket, we left a few runs out there", Williamson said. "They outplayed us today".
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She called for a medical timeout during the interval, before returning to the court with heavy bandage on her thigh. If Pliskova defeated Halep on Thursday, she would have automatically claimed the No. 1 spot in the WTA rankings.