Lokesh Rahul was batting on 18 with Murali Vijay on two at the other end as the hosts trailed Australia's first innings total by 431 runs.
Jadeja returned figures of 5/124, the eighth five-wicket haul of his Test career and the fourth this season.
Fast bowler Umesh Yadav also bowled well for figures of 3/106 while star off-spinner Ravichandran Ashwin bagged a wicket.
Skipper Steven Smith played a captain's innings for Australia and remained unbeaten on 178 to guide them to a big total. He hit 17 boundaries during the 361 balls he faced.
Glenn Maxwell, who returned to the Australian Test side after a lengthy period, scored his maiden century in the longest format of the game and put on a 191-run stand with Smith to help the visitors to a strong position after losing early wickets on the opening day.
The visitors started the day on a positive note with Maxwell bringing up the first runs of the day with a boundary.
The right-hander quickly completed his maiden century in the longest format of the game, reaching the landmark by guiding an Umesh Yadav delivery to the third man boundary.
Maxwewll, who is playing in only the fourth Test of his career, departed shortly afterwards when a Jadeja delivery landed on the off-stump line and spun away viciously. The ball kissed the outer edge of Maxwell's bat and Indian wicketkeeper Wriddhiman Saha pulled off an excellent catch.
Maxwell scored 104 runs off 185 balls with nine hits to the fence and two sixes. It was a rare display of patience and endurance from the right-hander who had scored only 80 runs in the three Tests he had played previously in his career.
Matthew Wade joined Smith in the middle and the duo proceeded to add 64 runs between them before Jadeja struck twice in quick succession.
Wade seemed to be in good form, scoring 37 runs off the 50 balls he faced. He seemed on track for a half-century but was outfoxed by a Jadeja delivery which kept straight after pitching.
Wade, who played for the turn, only ended up handing an edge to Saha.
Pat Cummins survived all of two deliveries before Jadaja bowled him with a beautiful delivery. The ball landed on the middle and leg stump line and spun away viciously, beating the outside edge of the bat to hit the off-stump.
Jadeja almost dismissed Smith in the last over before lunch, but the ball landed just short of the slip cordon.
India (First Innings): 20/0 (Lokesh Rahul 18 batting, Murali Vijay 2 batting) vs Australia (First Innings): 451 (Steven Smith 178 not out, Glenn Maxwell 104; Ravindra Jadeja 5/124, Umesh Yadav 3/106).
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