Tamim Iqbal (34) and Mushfiqur Rahim (35) took the hosts close to their target in an excellent morning session and they crossed the finish line on a deteriorating wicket shortly after lunch.
It was their first test victory since beating New Zealand in Mount Maunganui last year.
Ireland, with six debutants and playing a test match for the first time since 2019, made their highest first innings total of 214 on the opening day after opting to bat first as Harry Tector (23) top-scored with 50.
Bangladesh rode on Taijul Islam's (31) eleventh five-wicket haul to run through Ireland's line-up and posted a commanding 369 in response, thanks largely to Mushfiqur's superb 126 and skipper Shakib Al Hasan's (36) 87.
But the hosts were left frustrated after a defiant Ireland batted out the third day which they began at 27-4.
Tector was dropped on nine early on and made the most of the reprieve to hit 56, while fellow debutant Lorcan Tucker (26) smashed 108 before Ireland reached 286-8 at stumps to prolong the contest with a lead of 131.
Andy McBrine (29), who took 6-118 to dismiss Bangladesh and also impressed with the bat in Ireland's second essay, added only one more run to his overnight tally to fall for 72 early on the fourth morning as his team eventually folded for 292.
Ireland will play two tests in Sri Lanka later this month.
