Opener Matt Renshaw, who had retired due to upset stomach before lunch, was unbeaten on 38 at the break and giving him company was Mitchell Marsh (2).
Australia, who lost opener and vice captain David Warner in the pre-lunch session for 38, lost the wickets of captain Steven Smith (27), Shaun Marsh (16) and Peter Handscomb (22).
Spinners Ravichandran Ashwin (1/37) and Ravindra Jadeja (1/43) struck late blows to send back Smith and Handscomb respectively after Jayant Yadav outsmarted Shaun Marsh earlier in the second session.
But once they departed, the scoring rate dropped drastically. Skipper Smith cut out all frills and played a dour knock while Shaun was slightly more aggressive in making 16 before falling to Jayant.
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Handscomb was caught right in front of the wicket by Jadeja with a quicker, straight ball and five balls later Ashwin had Smith caught at mid-wicket coming down the track in a rush of blood.
India almost had next man Mitchell Marsh sent back caught behind off Ashwin in the same over in which he dismissed Smith, but a review of the decision revealed he had not nicked the turning ball.
Earlier, Warner was looking good before he was bowled for 38 by Umesh and then tall Renshaw retired to the pavilion with an upset stomach close to lunch, which resulted in the arrival of two new batsmen in captain Steven Smith and Shaun Marsh at the crease.
For India, the bulk of the bowling was done by their spin trump card Ashwin, who opened the attack with Ishant Sharma.
Umesh, brought into the attack after 27 overs, bowled two overs for one run and ended up with the wicket of Warner, who played away from his body and dragged the ball on to his stumps.
Opting to take first strike in the first-ever Test held in this city, Australia initially did struggle a bit against the turning ball.
They managed to score 40 runs in the first hour of play with Warner starting slowly and then attacking off-spinner Jayant Yadav.
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Ashwin, on the other hand, was not only turning the ball but also kept down the runs with Warner, especially, in no mood to try and attack the home team's spin trump card.
The Indians also wasted their first DRS call in the 10th over of the match after their appeal for a catch behind against the tall Renshaw was rejected by the umpire. The TV replays showed appreciable gap between the bat and the ball.
Warner, who was initially outscored by his opening partner, soon got past that phase by striking Jayant for a couple of fours in two overs, forcing India captain Virat Kohli to bring on third spinner Ravindra Jadeja.
The tall England-born Renshaw then got past his partner's score again as he first tickled a ball from Ashwin to fine leg for a boundary.
He then smacked Jadeja, who was not getting the same amount of purchase from the wicket as his spin mates, for a four and then lofted the left-arm spinner in the same over over long on for the first six of the match.
Warner, on 37, had two close calls. He first survived a big leg before shout from Ashwin and then nearly popped up a return catch to the off-spinner in the same over.
Warner's dismissal also saw Renshaw, not out on 36 off 89 balls, simultaneously retire to the pavilion with tummy trouble.