McCullum, who came in at first drop, smashed 123 off 58 balls, including 11 fours and seven sixes, as New Zealand posted an exceptional score of 3/191 off their 20 overs.
In reply, Bangladesh never looked likely, slumping to 4/37 with Kyle Mills ripping through the top order (3/33) before Nasir Hossain made the score respectable with 50 off 39 balls as they fell well short at 8/132.
New Zealand skipper Ross Taylor said: “When you have someone score a century in T20 you’re always in the box seat. The wicket nipped around early on and seemed to zip on a little bit under the lights. Hopefully this win will give us momentum.”
Bangladesh captain Mushfiqur Rahim acknowledged that McCullum was the difference with his big hitting.
“We bowled well in the first few overs but McCullum, even when he doesn’t hit the ball well, it’s still going for six. We’ll look at our strengths and come up with a new strategy for the next game,” he said.
McCullum’s 123 was the best ever score in T20 cricket, bettering the previous record jointly held by South Africa’s Richard Levi and West Indies’ Chris Gayle of 117.