Paceman Taskin Ahmed gave first breakthrough to Bangladesh as he dismissed Blackcaps opener Ronchi at 46 runs in 7.1 overs.
The Kiwis were 60 for 1 in 10 overs for a short while.
Earlier, the Kiwis captain Kane Williamson won the toss and elected to bat first in the match which was delayed due to wet outfield aftermath of rain.
New Zealand team: MJ Guptill, L Ronchi†, KS Williamson*, LRPL Taylor, NT Broom, JDS Neesham, CJ Anderson, MJ Santner, AF Milne, TG Southee, TA Boult.
Bangladesh team: Tamim Iqbal, Soumya Sarkar, Taskin Ahmed, Mushfiqur Rahim†, Shakib Al Hasan, Sabbir Rahman, Mahmudullah, Mosaddek Hossain, Mashrafe Mortaza*, Rubel Hossain, Mustafizur Rahman.