Pace bowler Mashrafe Mortaza proved to be the hero with bat and ball.
After taking 4-44 to limit the Black Caps to 224 all out, he then smashed two successive sixes off the hapless James Franklin to secure victory in the 49th over of an enthralling contest.
New Zealand came into the tournament as many peoples' dark horses for the title having swept world champions Australia 3-0 in a recent home series.
But they never managed to get to grips with either the clever pace of Mortaza or the left-arm spin of Abdur Razzak.
Off-spinner Daniel Vettori took three wickets for the Kiwis as he tried to put a brake on the scoring but Mortaza's big-hitting was to have the final say as Bangladesh celebrated their first ever win over New Zealand.
Earlier, New Zealand all-rounder Jacob Oram, who came into the World Cup nursing a broken ring finger on his left hand, batted through the pain barrier to rescue the Black Caps.
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Oram top-scored with 88 off 107 balls as New Zealand were all out for 224.