Vishwa Fernando struck twice after Zakir Hasan made a patient half-century as Sri Lanka put Bangladesh under pressure in the second test in Chittagong on Sunday.
Bangladesh were 115-4 at lunch on the third day, with left-arm spinner Prabath Jayasuriya also chipping in with a wicket in the morning session.
Veteran batsmen Mominul Haque, on two, and Shakib Al Hasan, who had made six, were batting at the break with Bangladesh still trailing Sri Lanka by 416 runs.
Zakir survived some early scares to make 54, his fourth test fifty, before Sri Lanka broke through.
Resuming on 55-1 with nightwatchman Taijul Islam, the left-hander saw a catch nearly carry to Kamindu Mendis at slip before a leg-before appeal was turned down.
Sri Lanka ended Bangladesh's dogged resistance when Fernando, a left-arm quick, swung one in to uproot the leg stump of Zakir.
Skipper Najmul Hossain Shanto looked edgy from the start and faced 11 balls for his one as he departed after attempting an on drive, which was caught at short midwicket by Dimuth Karunaratne.
Fernando bowled Taijul for 22 with another inswinging delivery to reduce Bangladesh to 105-4.
Sri Lanka dominated Bangladesh in the first two days to amass 531, the highest test total without any batsman scoring a century.
BANGLADESH: Zakir Hasan, Mahmudul Hasan Joy, Najmul Hossain Shanto (capt), Mominul Haque, Shakib Al Hasan, Shahadat Hossain, Liton Das, Mehidy Hasan Miraz, Taijul Islam, Syed Khaled Ahmed, Hasan Mahmud.
SRI LANKA: Dimuth Karunaratne, Nishan Madushka, Kusal Mendis, Angelo Mathews, Dinesh Chandimal, Dhananjaya de Silva (capt), Kamindu Mendis, Prabath Jayasuriya, Asitha Fernando, Vishwa Fernando, Lahiru Kumara.