India had the slight edge off-spinner R Ashwin cleaned up the Australian top-order and a meticulous pace effort kept the home team's scoring in check on the second day.
Travis Head (61 batting) and Pat Cummins (10) kept Australia in the game with a 50-run partnership in the last session but India walked away with the advantage after the latter fell towards the end of day's play.
At stumps, Australia were 191/7 in 88 overs with Ashwin being their primary tormentor, securing figures of 3/50 in 33 overs. He was aided well by the pace duo of Jasprit Bumrah (2/34) and Ishant Sharma (2/31).
Post-tea, Peter Handscomb (34) had been living dangerously and he was the first to fall. He tried to play the late cut against Bumrah but only managed to edge behind.
Travis Head started to play more shots as soon as the tail-ender came to the crease, and in doing so, reached his second Test half-century off 103 balls.
In the morning, Australia were placed at 57-2 at lunch after India were bowled out for 250 (88 overs) on the very first ball of the morning session.