Electing to bat first, Bangladesh managed a below-par 141/8 riding on a gritty half-century from Mahmudullah at the McLean Park, a news agency reported
The tourists once again put up a poor show with the bat as their top order cheaply surrendered before Mahmudullah showed some resolve.
Captain Mashrafe Bin Mortaza and his men then staged a good fight early in New Zealand's chase. They raised Bangladesh hopes by reducing the hosts to 62/4 in the 11th over.
But Williamson (73 off 55 balls) then paired with Colin de Grandhomme (41 off 22 balls) for an unbroken 81-run partnerhip off just 47 deliveries, to coast to the win with two overs to spare.
While Williamson and de Grandhomme did appear to be in control as the overs ticked down, the chase looked nervy until they took 17 runs from the 16th over bowled by Soumya Sarkar.
Williamson smashed five boundaries and two sixes on his way to scoring his sixth T20I half-century, in the process becoming the fourth New Zealand batsman to score 1,000 T20I runs.
De Grandhomme's cameo contained three fours and as many sixes.
Rubel Hossain, Shakib Al Hasan and Mustafizur Rahman each picked up a wicket for Bangladesh.
Earlier, Mahmudullah returned to form by cracking a gritty half-century.
But Bangladesh were put the sword early on by debutant pacers Lockie Ferguson (3/32) and Ben Wheeler (2/22) who reduced them to 30/4.
After opener Imrul Kayes was dismissed for a duck Tamim Iqbal (11), Sabbir Rahman, Shakib got starts but could not stretch their innings.
Soumya Sarkar continued his poor form and got out for a golden duck.
Mahmudullah then dug deep and struck a 47-ball 52, smashing three boundaries and as many sixes to help the Tigers post a respectable total on the scoreboard.
Mosaddek Hossain (20 off 17 balls) also played a good hand down the order.
But Bangladesh, who have never beaten New Zealand in a T20, were left ruing the fact that 30 runs more from the top order on a good paced wicket with short square boundaries may have changed things in New Zealand's chase.
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.
The teams now head to Mt Maunganui for the remaining two games on January 6 and January 8.
Bangladesh: 141/8 in 20 overs (Mahmudullah 53; Lockie Ferguson 3/32)
New Zealand: 143/4 in 18 overs (Kane Williamson 73 n.o., Colin De Grandhomme 41 n.o; Shakib Al Hasan 1/30, Rubel Hossain 1/43, Mustafizur Rahman 1/21)