Set a target of 248, New Zealand caved in for 207 in 46.4 overs to suffer their sixth successive ODI defeat to Bangladesh, who won Tuesday's series opener by 43 runs.
Off-spinner Sohag Gazi and paceman Mashrafe Mortaza led Bangladesh's bowling attack, sharing six wickets between them to stifle New Zealand, who lost to Bangladesh 4-0 in their previous visit in 2010.
Mashrafe provided the early breakthrough when he bowled Hamish Rutherford for one before Sohag took a return catch to dismiss the other opener, Anton Devcich for 19.
Corey Anderson (37) and Ross Taylor tried to repair the damage, but a stunning catch by Mushfiqur Rahim off Mashrafe ended their 61-run stand and gave Bangladesh a firm grip on the match.
Bangladesh sensed victory when occasional left-arm spinner Mominul Haque had New Zealand captain Brendon McCullum out lbw for 14.
Taylor (45) held down one end before he was caught by Mahmudullah off Sohag at long on.
Some late blows by Nathan McCullum (25 off 23 balls) and unbeaten Kyle Mills (27 off 25 balls) only delayed the inevitable.
Anderson finished with 4-40 while Neesham claimed 4-53, his second successive four-wicket haul in the series.
Bangladesh also lost wickets at crucial times even though most of their batsman got a decent start.
Tamim Iqbal scored an arduous half-century but was out just when the team were looking for him to accelerate.
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