Kuldeep Yadav, playing his first Test of the series, scalped three wickets, while Ravindra Jadeja took two - this after Australia had made their most assured start of the summer, and were comfortably placed at 128/1.
India, having failed to win a Test series in Australia in 11 previous attempts, will end a slump that dates back to when they were crushed by Don Bradman's side in 1948.
Australia haven't completed a four-Test series at home without at least one player reaching three figures.
England were the last team to force Australia to follow-on at home in the drawn 1988 Sydney Test, a result the hosts are unlikely to repeat unless the weather intervenes decisively.
The fourth day of the Test started nearly four hours late due to rain, with the home team resuming on 236 for six in reply to India's thumping first innings 622 for seven declared. Pacer Mohammed Shami also grabbed a couple of wickets while Jasprit Bumrah managed a wicket as well in the first innings.
The recalled Peter Handscomb began with goal, confidently stroking two boundaries to move to 37 before swiping at a Bumrah delivery and dragging it onto his stumps.
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Kuldeep Yadav then dismissed Nathan Lyon for 0.
Bad light and intermittent rain led to substantial loss of play with only 25.2 overs bowled on day four.
Josh Hazlewood was dropped by Hanuma Vihari off Kuldeep's bowling when he had yet to score and he and fellow tailender Mitchell Starc (29 not out) fought a dogged rearguard to put on 42 for the final wicket.
Yadav got his reward thereafter with Hazlewood trapped lbw on a close umpire's call, and the decision stayed in the wrist-spinner's favour despite DRS.
Australia, resuming at 236-6, quickly lost 3-22 a day after their top order folded in a collapse of 3-24.