At the break, Peter Handscomb and Mitchell Starc were faced with an onerous task of negotiating a determined Indian attack on a difficult pitch at the M Chinnaswamy Stadium.
The visitors were another 87 runs from the target with four wickets in hands to retain the trophy.
Leading the four-match series 1-0, Australia had bowled out India for 274 in the second innings with pacer Josh Hazlewood (6/67) doing the bulk of damage in the first session.
The Indian bowlers, led by Ravichandran Ashwin and Umesh Yadav, rattled the Australian top-order.
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Ashwin, who had just one wicket in the first innings, picked up three for 35, including those of David Warner (17), Mitchell Marsh (13) and Mathew Wade (0).
Ashwin has so far got Warner nine times in 10 matches.
Umesh (2/26) struck in first over of his second spell when he removed Shaun Marsh (9) leg before wicket.
The pacer then, in his fifth over, sent back Aussie captain Steve Smith (28) leg before wicket, which was followed by a bit of drama. Smith first turned to the non-striker's end to seek an opinion and sparks flew when the umpires noticed that the batsman had turned towards the dressing room as well.
Ishant Sharma (1/28) gave India the first breakthrough when he sent back Mathew Renshaw for five. The left-hander, who had scored a dogged 60 runs in the first innings, fell cheaply for 4, leaving Australia at 22 for one.
Ashwin bagged his first wicket when he trapped Warner leg before, who reviewed it but the third umpire's decision went against him.
Umesh then removed Shaun Marsh, who did not offer a shot to leave Australia struggling at 67 for three. The next three wickets fell for 17 runs.
Hazlewood (6/67) and Starc (2/74) were the wrecker-in-chiefs in the first session, cutting short India's second innings in 97.1 overs.
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Hazlewood bowled his heart out with the second new ball, which was taken after 80 overs. Trailing 0-1 in the four-match series, India were overnight 213 for four with Cheteshwar Pujara, who would eventually fall for 92, batting on 79.
Hazlewood took the prize wickets of Abhinav Mukund, skipper Kohli and Ravindra Jadeja yesterday and dismissed Pujara, Ravichandran Ashwin (4) and Umesh Yadav (1) in the fourth morning.
The last wicket was taken by Stephen O'Keefe when he removed Ishant Sharma for six off 28 balls, ending a dogged 16-run partnership.
Pujara and Rahane added 25 runs this morning before Rahane was trapped in front of the wicket by Starc.
Local lad Karun Nair, who had scored a brilliant triple century against England, failed for the second time in front of his home crowd as he was cleaned up by Starc.
In the same over, Ashwin was cleaned up by Hazlewood, and Ishant Sharma and Umesh Yadav did not survive long.