Having let India off the hook after dominating the first two sessions on Thursday, Bangladesh were reeling at 26-3 at lunch on the second day of the contest at the M.A. Chidambaram Stadium.
Skipper Najmul Hossain Shanto (15) and Mushfiqur Rahim (four) were in the middle with Bangladesh 350 behind.
Jasprit Bumrah struck in the first over of the Bangladesh innings when Shadman Islam shouldered arms to a delivery that went on to hit the top of the off stump.
Akash Deep bowled Zakir Hasan and Mominul Haque with successive deliveries but was denied a hat-trick by Mushfiqur.
Earlier, Ravindra Jadeja fell for his overnight score of 86 after India resumed on 339-6.
While Jadeja fell short of a century, his 199-run stand with Ravichandran Ashwin (113) rescued India from a top-order meltdown.
Hasan Mahmud (5-83) was pick of the Bangladesh bowlers, while Taskin Ahmed (3-55) ran through the Indian tail.