India reached 40-0 at stumps, needing another 152 to win their third straight match and the five-Test series.
Skipper Rohit Sharma was unbeaten on 24, with Yashasvi Jaiswal on 16.
Opener Zak Crawley earlier top-scored with 60 but the rest of the England batting faltered against India's spin, with Ashwin returning figures of 5-51 and Kuldeep Yadav 4-22.
England lost regular wickets to end their innings in the final session, leaving it up to the bowlers to level the series and send it to a fifth Test decider in Dharamsala.
India's bowlers did their job after wicketkeeper Dhruv Jurel hit 90 in the morning session to reduce the hosts' deficit after they began the day on 219-7.
Ashwin took the new ball on a pitch with cracks and variable bounce and struck in consecutive balls to remove Ben Duckett, caught at short leg for 15, and Ollie Pope, lbw without scoring, as England were reduced 19-2.
Root, who scored an unbeaten 122 in the first innings, put on 46 with Crawley before Ashwin and India reviewed an lbw appeal, with tracking software indicating the ball pitched just in line and would have hit Root's leg stump.
Crawley raised his fifty but was bowled by Yadav, who turned one sharply with his left-arm wrist spin.
Yadav then removed Ben Stokes just before tea when the England captain was undone for the second time in the match by a ball that barely bounced ankle high and was bowled off his pad.
Jonny Bairstow fell on the first ball after tea and the innings folded soon after. Ben Foakes fell for a 76-ball 17 off Ashwin.
"I've enjoyed bowling with the new ball and today was one of those days where Rohit asked in the huddle who wants to start and I put my hand up," said Ashwin, who opened the attack for India, while also heaping praise on Kuldeep.
"We all know how many revs he can put on the ball and what skills he's got, but when you can change up the pace and the trajectory, he's double the bowler he is."
England spinner Shoaib Bashir took his first Test five-wicket haul to help bowl India out for 307, giving the tourists a lead of 46 going into their second innings.
India were rescued in the morning session by Jurel, who put on a 76-run eighth-wicket stand with overnight partner Yadav.
Veteran fast bowler James Anderson broke the stand when Yadav, on 28, played on to give the veteran seamer his 698th Test wicket.
The 41-year-old is now two victims away from becoming just the third bowler to take 700 wickets in Tests, after Sri Lanka's Muttiah Muralitharan (800) and the late Australian great Shane Warne (708).
Jurel, who made his debut in the third Test in Rajkot, hit six fours and four sixes in his 149-ball stay and picked up his maiden Test half-century.
Bashir, who missed England's win in the first Test due to a visa issue, trapped debutant Akash Deep lbw for his fifth wicket and kissed the ground in celebration.
