Bangladesh were bowled out for 232 in 49th over as Mithun's 62 off 90 rescues Tigers in the Bangladesh tour of New Zealand at McLean Park, Napier.
Boult and Santner picked up three wickets while the visitors were reeling at 94/6 before Mohammad Mithun's third ODI fifty kick-started a grand recovery against the host.
Mohammad Mithun scored 62 runs off 90 balls smashing five fours, rescuing Tigers from the deep danger in the soil of New Zealand.
Earlier, Bangladesh won the toss and opted to bat first as they chase a maiden victory over the Black Caps in the first one-day international in Napier.
The Bangladesh have never beaten the hosts in any format during four previous tours to New Zealand.
Bangladesh Team: Tamim Iqbal, Liton Das, Soumya Sarkar, Mushfiqur Rahim, Mohammad Mithun, Mahmudullah Riyad, Sabbir Rahman, Mohammad Saifuddin, Mehidy Hasan, Mashrafe Mortaza (capt), Mustafiqur Rahman
New Zealand Team: Martin Guptill, Henry Nicholls, Kane Williamson (capt), Ross Taylor, Tom Latham, Colin de Grandhomme, Jimmy Neesham, Mitchell Santner, Matt Henry, Lockie Ferguson, Trent Boult.