The touring side, trailing 2-0 in the five-match series, had been cruising on 230-2 thanks largely to left-handed opener Jennings who made 112 after being dropped on nought.
But Ashwin, the world's top-ranked test bowler, picked up four wickets to peg back England who had been eyeing a large first innings total on a track already assisting the spinners.
At stumps honours were even with Ben Stokes unbeaten on 25 and Jos Buttler on 18.
Jennings, the son of South African former wicketkeeper and coach Ray, only arrived in India on Monday after being called up from the England Lions squad in the United Arab Emirates.
After skipper Alastair Cook won the toss and opted to bat, the 24-year-old Jennings was dropped by Karun Nair at gully off paceman Umesh Yadav. India also wasted a review after Jennings was given not out to an appeal for leg before off Bhuvneshwar Kumar, who came into the side for the injured Mohammed Shami.
He settled down though and grew in confidence as he shared in an opening stand of 99 with Cook.
Cook looked solid during his 60-ball 46 and hit five boundaries before he was out to left-arm spinner Ravindra Jadeja's third ball of the morning.
Dancing down the wicket he was beaten by the spin and was stumped by wicketkeeper Parthiv Patel.
Jennings played some handsome drives against the pacemen and some fine sweep shots to the spinners. He reverse swept off-spinner Jayant Yadav for a boundary to bring up his hundred.
England lost Joe Root for 21 after lunch when the right-hander edged Ashwin to India captain Virat Kohli at slip.
But Jennings and Moeen Ali kept England on course by stitching together a stand of 94 for the third wicket.
Kohli made a valiant effort and got his fingertips to an edge from Moeen, then on 13, off Umesh but failed to hang on.
Kohli then turned to Ashwin to break the stand and the off-spinner obliged by picking up Moeen and Jennings in three balls in the first over of his new spell.
Moeen fell on 50, top-edging a sweep, while Jennings got an edge to second slip to end an innings that contained 13 fours.
Ashwin also dismissed Jonny Bairstow for 14 as England stalled, losing three wickets for the addition of 19 runs.
In an unfortunate incident in the second session, Australian umpire Paul Reiffel left the field after being hit on the back of his head by a throw from the outfield but was givem the all clear after a visit to the hospital for scans.
(Editing by Amlan Chakraborty)
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