Boult broke a resolute 235-run stand between Soumya Sarkar and stand-in captain Mahmudullah that frustrated New Zealand's bowlers as they toiled in warm conditions at Seddon Park.
The left-arm seamer got the second new ball to swing back, brush Sarkar's pad and cannon into his off-stump to end the 26-year-old's 171-ball innings on 149. It was Sarkar's first test century.
The breakthrough effectively ended Bangladesh's resistance, even if Mahmudullah did his best to protect the tail, and they were eventually dismissed for 429 about 20 minutes after tea.
Mahmudullah made 146 and was caught on the boundary by Boult, who finished with figures of 5-123, before Tim Southee had Ebadot Hossain caught by BJ Watling, who broke Adam Parore's New Zealand record for test dismissals by a wicketkeeper.
New Zealand had scored 715-6 declared, with captain Kane Williamson making 200 not out, while openers Tom Latham and Jeet Raval also scored centuries.
The visitors had been in danger of being dismissed inside three days when they were reduced to 126-4 after tea on Saturday before Sarkar combined with Mahmudullah.
The pair managed to take the visitors through to 174-4 at the close of play and continued to thwart the hosts' bowlers during Sunday's first session with Sarkar going from 39 to his century, bringing it up with a single off Southee.
They took their side to 322-4 at lunch and continued to resist New Zealand after the break, even when they took the second new ball.
Boult, however, managed to dismiss Sarkar and then bowl Liton Das for one and Khaled Ahmed for three before Southee had Mahmudullah caught in the deep after tea and then ended the match when Hossain nicked behind.
The second match of the three-test series starts in Wellington on Friday.
(Reporting by Greg Stutchbury; Editing by Sudipto Ganguly)
