Tamim Iqbal (seven) and Liton Das (17) came on to set up Bangladesh’s chase and went 23-0 in three overs.
On the previous day, Ireland lost just four wickets in three sessions to post 259 runs, leading Bangladesh by 131 runs at stumps.
Ireland’s recovery, led by a maiden Test century from Lorcan Tucker (108) and 56 from Harry Tector, was reduced to 27–4 at stumps on the second day.
On the fourth day, spinners Taijul and Mehidy Hasan Miraj opened the proceedings for the Tigers.