The left-hander top-scored with 63 to help India surpass Australia's first innings total of 300 and then snared three wickets with his spin bowling to dim the visitors' hopes of victory.
India's pacemen did the early damage. Umesh Yadav picked up two wickets and Bhuvneshwar Kumar dismissed the in-form Australian skipper Steve Smith for 17 to leave the Aussies reeling on 31-3 inside 10 overs.
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A short ball hit the bottom edge of Smith's bat before crashing onto the stumps in spectacular fashion.
Smith amassed 499 runs including three centuries from the four Tests, only the sixth visiting batsman to score so many hundreds in a series in India.
The spinners then got down to business on a lively track that offered plenty of pace and bounce, with Jadeja and Ravichandran Ashwin grabbing three wickets each to skittle out Australia for 137 in the final session of the day.
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At stumps the top-ranked home side were 19-0, needing just 87 more to win the match and clinch the Border-Gavaskar trophy with a 2-1 series scoreline.
With two days of play remaining and all their wickets intact, India look favourites to seal a memorable win, capping weeks of controversies and verbal volleys between the top sides.
India got their first breakthrough when Umesh Yadav had David Warner caught behind cheaply for six, just after the opener was dropped in the slips by the butter-fingered Karun Nair.
Nair had dropped the same batsman in the first innings as well.
Warner ended the four-Test series with an average of 24.12 — his lowest in a series of four or more matches, and his third lowest overall.
The Indian pace bowlers exploited the wicket to put relentless pressure on the batsmen.
Yadav also accounted for Matt Renshaw before Glenn Maxwell put on 56 for the fourth wicket with Peter Handscomb to steady the ship briefly.
But Handscomb was brilliantly caught by India's stand-in skipper Ajinkya Rahane off Ashwin while Jadeja dismissed Shaun Marsh for one.
Jadeja had played a crucial knock with the bat earlier in the day, starring in a seventh-wicket stand of 96 with Wriddhiman Saha (31) to help India post 332 and swing the momentum their way.
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