Henry Nicholls also stroked his way to 107 - his fifth test century - as the hosts declared at 432 for six for a 221-run first innings lead after tea on the fourth day.
A courageous innings of 67 by captain Mahmudullah delayed New Zealand's push for victory and it was forced to claim an extra 15 minutes after the scheduled lunch break to take the last two wickets it needed to seal the match and series.
Neil Wagner finished with figures of 5-45.
Boult was the most successful bowler with two wickets, while Henry claimed one.
New Zealand skipper Kane Williamson, who suffered a shoulder injury on the third day of the ongoing second Test against Bangladesh was taken for scans on Monday.
And they quickly made inroads, reducing Bangladesh to 80-3 at stumps, the tourists still trailing by 141 runs and facing defeat in a Test in which the first two days were washed out.
Kane Williamson was captain of the franchise previous year and led his team to IPL finals.
Captain Kane Williamson was absent in the Bangladesh second innings as he underwent an MRI scan on his shoulder injury.
"But he's also aware of it's not quite 100 then we will potentially hold him back and just make sure he is right before he goes on that plane".
After completing the double century, Taylor was dismissed by Mustafizur Rahman in the following over when he tried a big hit. "We didn't see it unfolding as it did".
Then it becomes slightly tricky for Stead and New Zealand Cricket.
Head coach Gary Stead said Williamson will travel with the team to Christchurch where he'll be further assessed ahead of the third and final Test, starting on Saturday at Hagley Oval.
New Zealand won the first test by an innings and 52 runs. While their ground fielding was not good enough, they twice dropped Ross Taylor, who later went on to score 200 to throw Bangladesh out of the match.
Only captain Mahmudullah Riyad and Mohammed Mithun, who scored 67 and 47 respectively, offered serious resistance.
"We know that there was a bit of help with the new ball and even with the semi new ball but we should have applied ourselves a bit better, so going forward to the next match we need to adapt as soon as we can".