Mahmudullah, who failed to fire in the one day series with scores of 0, 1 and 3, held together the visitors' below-par innings of 141 for eight with his third Twenty20 international half-century.
While Williamson and de Grandhomme looked in control of achieving the target as the overs ticked down, they did not really seal it until they took 17 runs from the 16th over bowled by Soumya Sarkar.
De Grandhomme then ended the game with a six over extra cover on the final ball of the 18th over to take New Zealand to 143 for four.
Bangladesh, who have never beaten New Zealand in a Twenty20 match, would be ruing the fact they were probably 30 runs short on a good paced wicket with short square boundaries.
Lockie Ferguson, making his international Twenty20 debut, had taken two wickets in his first two balls but was denied the hat-trick when Mahmudullah clamped down on the 147 kph inswinging yorker.
He eventually fooled the right hander in the final over with a slower delivery to finish with figures of three for 32.
The two sides now head to Mt. Maunganui for the remaining two games on Jan. 6 and Jan. 8.
(Reporting by Greg Stutchbury in Wellington; Editing by Amlan Chakraborty)
