Bangladesh seamer Hasan Mahmud's three-wicket burst in the morning session had India reeling at 34-3 inside the opening hour at the M.A. Chidambaram Stadium.
Opener Yashasvi Jaiswal made 56 but it was the marathon 195-run unbroken stand for the seventh wicket between Ashwin and Jadeja that helped India claw their way back into the contest.
Ashwin's unbeaten 102 off 112 balls, his sixth Test hundred, included 10 fours and two sixes. Jadeja, batting on 86, was closing in on his fifth Test hundred.
"On a surface like this with a bit of spice, if you're going after the ball, might as well go after it really hard," Ashwin said.
"It's the old Chennai surface with a bit of bounce and carry and the red soil pitch allows you to play a few shots.
"If you're willing to just get in line... and can give the ball a little bit of tonk, it really helps."
Bangladesh are on a high after their 2-0 series sweep in Pakistan earlier this month and they dominated the first two sessions after captain Najmul Hossain Shanto elected to field.
Home captain Rohit Sharma (six) needed 11 balls to get off the mark and survived a vociferous lbw appeal before edging Hasan to Shanto at second slip.
In his next over, Hasan had Shubman Gill caught behind for a duck tickling the ball down the leg side.
Virat Kohli also fell to a soft dismissal, driving away from his body to be caught behind for six.
Challenging conditions and departing partners could not bog down Jaiswal though and the opener played gorgeous drives and exquisite flicks.
Rishabh Pant (39), playing his first Test since a horrific car crash in December 2022, cut and pulled fearlessly and scored briskly to turn the heat back on the bowlers.
Hasan broke the 62-run stand soon after the lunch break when Pant's lazy waft to a delivery wide outside the off-stump led to another soft dismissal.
Nahid Rana ended Jaiswal's fluent knock and spinner Mehidy Hasan Miraz removed KL Rahul for 16 before local boy Ashwin combined with Jadeja to launch a spectacular counter-attack.
