Jasprit Bumrah had Australia's batsmen tied up in knots with his immaculate line and length, giving India the decisive advantage that withstood a 2nd innings batting collapse in the 3rd Test.
At the end of day three, India, after bowling out Australia for 151 in response to their first innings total of 443/7 declared, took a 346-run lead.
In their second innings, India were jolted by Pat Cummins (4-10), who took four wickets for two runs in the space of 19 balls.
Mayank Agarwal (28 *), who scored an impressive 76 in the first innings, and Rishabh Pant (6) were at the crease after 15 wickets fell in all on Day 3.
India were reduced to 32-4 with a double break against Kohli and Rahane. Rohit Sharma was caught at slip off Josh Hazlewood (1-13).
The four-match series is level at 1-1, after India won the first Test in Adelaide by 31 runs and Australia won the second Test in Perth by 146 runs.