Resuming on an overnight score of 272-6 in their second innings, Bangladesh soon lost the wicket of Mehidy Hasan Miraz (13) to Mohammed Siraj.
Captain Shakib Al Hasan (84) was next to depart as his entertaining display, which featured six fours and six sixes, came to an end after Kuldeep Yadav found the middle stump as Shakib tried an aerial stroke.
Kuldeep then dismissed Ebadot Hossain for a duck, before Axar Patel bowled out Taijul Islam for the same score, finishing with figures of 4-77 in the second innings.