Chasing a modest target of 245, Ross Taylor and Kane Williamson shared a third-wicket partnership of 105 to put the Black Caps on course for a comfortable victory, but they lost six more wickets before notching up their second win of the tournament with 17 balls to spare.
Williamson should have been run out for eight but Bangladesh wicketkeeper Mushfiqur Rahim dislodged the bails with his arm to waste the chance and the New Zealand captain went on to make 40.
Tom Latham was out cheaply and when Taylor was caught by Mushfiqur for 82 Bangladesh scented a famous win.
Colin de Grandhomme, Jimmy Neesham and Matt Henry also fell as the pressure mounted but Mitchell Santner and Lockie Ferguson stayed calm to edge New Zealand over the line.
Shakib Al Hasan top-scored for Bangladesh with 64 and several batsmen made starts but they lost wickets at regular intervals and were never able to dominate a disciplined bowling attack.
(Reporting by Ed Osmond, editing by Pritha Sarkar)
