The Tigers began the day trailing India by 343 runs after the hosts declared on their overnight score of 493-6.
But the visitors were skittled out for just 213 as seamer Mohammed Shami and off-spinner R Ashwin ran through the batting line up with only Mushfiqur Rahim (63) standing in the way.
Mushfiqur made the most of a reprieve earlier in his knock to be 53 not out, having added 56 runs with Mehidy Hasan Miraz 38.
Bangladesh's efforts to stave off defeat were dealt an early blow when opener Imrul Kayes (6) was bowled by seamer Umesh Yadav with the score at 10.
Shadman Islam (6) fell shortly afterwards as a length ball by Ishant Sharma sneaked through his defence and crashed into the stumps.
Captain Mominul Haque was the next to go after being trapped lbw by Mohammad Shami for 7.
Mohammad Mithun followed the skipper back to the pavilion after a quick-fire 18, including four boundaries, when he miscued a short-ball by Shami straight to Mayank Agarwal at mid-wicket.
A 63-run partnership between Mushfiqur Rahim and Liton Das offered some encouragement for Bangladesh supporters on an otherwise miserable day for the tourists.
But hopes of a fightback were nipped in the bud when R Ashwin removed Liton for 35.
Kolkata hosts the second and final match, which will be the first day-night test for both sides, from Nov 22.
