England were reeling at 103 for five at stumps in response to India's first innings score of 455. The visitors still trail by 352 runs.
Ashwin, who scored a gritty half-century to propel the Indian score after skipper Virat Kohli's 167, claimed two wickets in the final session to rattle the English top-order.
Pace spearhead Mohammed Shami struck early to shatter skipper Alastair Cook's off stump in an early blow as England began their first innings reply before tea.
Shami bowled an inspired first spell with the new ball as he cleaned up Cook for two in the third over of the innings.
Shami's sensational delivery pitched on a good length before moving in a shade to hit the top of Cook's off stump and break it in two, leaving England 4-1.
Haseeb Hameed, who impressed on debut with a well-made 82 in the second innings of the Rajkot Test, put on 47 runs with Joe Root (53) in his bid to steady the innings.
But Hameed's resistance did not last long as he was run out for 13 after being sent back while attempting to take a second run.
Off-spinner Ashwin soon took charge to get Ben Duckett clean bowled for five and then got the big wicket of Root, who recorded his 24th Test half-century.
Root tried to counter the Indian spin attack during his 98-ball stay laced with six boundaries, but ran out of patience after giving away a catch to long-off.
Debutant off-spinner Jayant Yadav joined his senior partner from the other end to get his first Test wicket after trapping Moeen Ali lbw for one as England slipped to 80-5.
Stokes and Bairstow then played cautiously to see off the rest of the day's play with a 23-run unbeaten stand.
Earlier Ashwin, who scored 70 in the Rajkot Test, continued his good run of form with the bat before the hosts were bowled out in the afternoon session.
Ashwin's fighting 58 came after off-spinner Ali checked the Indian charge with three wickets in the morning session, including Kohli, who added just 16 runs to his overnight 151.
Ashwin put on 64 runs for the eighth wicket with Jayant Yadav, who scored 35, guiding India to a respectable first innings score on a pitch that is expected to offer turn as the match progresses.
Kohli's 226-run third-wicket partnership with Cheteshwar Pujara (119) on the opening day was the highlight of the Indian innings after the hosts elected to bat first.
Original source: Ashwin puts India in control of second Test