The hosts started Day 4 on 212-3 but lost Shanto in the second over when Tim Southee had him caught down the leg side, ending a 98-run fourth-wicket stand.
Mushfiqur Rahim continued and raised his 27th test half-century off 79 balls, with an outside-edge two against Southee. But he failed to convert it to a hundred when left-arm spinner Ajaz Patel trapped him in front for 67.
Leg-spinner Ish Sodhi got the better of Shahadat Hossain (18) to expose Bangladesh's lower order.
Nurul Hasan (10) played a chancy innings before he was caught and bowled by Glenn Phillips, giving the offspinner his first wicket in the second innings.
Patel, who returned 2-137, could have also dismissed Mehidy Hasan had Daryl Mitchel not put down a catch at slip.
Mehidy continued and hit Phillips for a boundary to take Bangladesh's lead past 300 in the second innings with five sessions remaining. He was unbeaten on 32 at the interval. (AP) APA APA