After Pat Cummins (4-29) and a brilliant fielding display saw India dismissed for 244 in response to Australia's first-innings 338, the hosts reached 103-2 at stumps – a lead of 197 runs.
Cheteshwar Pujara (50) led India's batting effort, but they lost 49-6 to finish their innings as the SCG pitch began to show signs of uneven bounce.
Making matters worse for the tourists, Rishabh Pant (left elbow) and Ravindra Jadeja (left thumb) were taken for scans after suffering blows while batting.
Marnus Labuschagne (47) and Steve Smith (29), who led Australia's first-innings batting effort, were unbeaten at stumps as the hosts put themselves in position to take a 2-1 series lead.
It came after some wonderful fielding and a fine bowling display by Cummins reduced India.
Cummins had Ajinkya Rahane (22) playing on for the first breakthrough of day three before Josh Hazlewood (2-43) stepped up in the field.
The Australia paceman moved to his right at mid-off before quickly unleashing a throw while diving to run out Hanuma Vihari (4).
Pant responded for India but he struggled after a blow to his elbow and was caught at first slip by David Warner off Hazlewood.
Cummins had the dangerous Pujara caught behind, while Ravichandran Ashwin (10) was run out and Labuschagne produced a wonderful direct hit to also run out Jasprit Bumrah (0).
Jadeja (28 not out) played some nice shots and India were boosted by a couple of quick wickets to begin Australia's innings.
Mohammed Siraj (1-20) had Will Pucovski (10) caught behind and Warner (13) was trapped lbw by Ashwin (1-28), but Smith and Labuschagne steadied the innings to ensure Australia stayed in control.