Fast bowler Mustafizur Rahman claimed three wickets as Bangladesh skittled out New Zealand for their joint-lowest Twenty20 total of 60 in the first of the five matches on Wednesday.
Skipper Tom Latham and Henry Nicholls scored 18 each before New Zealand were bowled out in 16.5 overs following their decision to bat first in Dhaka.
The Kiwis were dismissed for the same score against Sri Lanka in 2014 at Chattogram.
Spinners Nasum Ahmed, Shakib Al Hasan and pace bowler Mohammad Saifuddin took two wickets each.
Mahedi Hasan struck in the first over with debutant Rachin Ravindra out for a golden duck after being caught and bowled by the off-spinner.
Shakib bowled Will Young for five and left-arm spinner Nasum then got two wickets in one over including Colin de Grandhomme, for one, to reduce New Zealand to 9-4.
Latham and Nicholls put on 34 runs for the fifth wicket but Saifuddin broke the stand with the captain's wicket to rattle New Zealand further.
Debutant Cole McConchie was out for nought and Saifuddin sent back Nicholls as the New Zealand batting fell like a pack of cards.
Bangladesh, who bowled out Australia for 62 last month to seal the T20 series 4-1 in Dhaka, need 61 to beat New Zealand for the first time in T20 internationals.