India declared at 106-8 after resuming on 54-5 on Saturday, setting Australia a massive target. India has five more sessions in which to claim the eight wickets required for victory and a 2-1 lead in the four-match series, on a pitch that is troubling batsmen with its variable bounce and turn.
After showing little resistance in a first-innings tally of 151, Australia began its second innings in similar style as openers Aaron Finch (three) and Marcus Harris (13) fell cheaply.
Finch was caught at second slip in the second over, after trying to cut a ball that was too close to him off Jasprit Bumrah's (1-12) bowling. One-day captain Finch's test spot appears under threat after scoring 11 runs in the third test for a total of 97 at 16.16 in the current series.
Harris was caught at short leg off the bowling of spinner Ravindra Jadeja (1-20) to leave Australia in trouble at 33-2.
Usman Khawaja, dropped on zero, is on 26 and Shaun Marsh is on two.
In a lively start to the fourth day's play, Australia quick Pat Cummins capitalized on the controversial pitch's variable bounce as he claimed a career-best 6-27 off 11 overs, including two wickets in the first hour of play.
Resuming on 54-5, India's was determined to move rapidly towards a lead of around 400, as Mayank Agarwal (42) and Rishabh Pant (33 off 43 balls) both cleared the rope.
Agarwal was bowled by the in-form Cummins with a delivery that kept low, and Ravindra Jadeja (five) edged a ferocious bouncer from Cummins to gully at 100-7.
It was the third time in 17 tests that 25-year-old Cummins had claimed a five-wicket haul, in an injury-interrupted career which started in 2011 at the age of 18.
Australia was skittled in dramatic fashion for 151 in its first innings on Friday in reply to India's 443-7 declared.
Fast bowler Bumrah, in his first year in test cricket, took a career-best 6-33 as India claimed a 292-run lead on the first innings.
India has never won a test series in Australia.