The hosts resumed on 236-6 in reply to India's 622-7 but lost Pat Cummins to the sixth ball of day four, and Peter Handscomb soon followed for 37.
The tail offered some resistance but left-arm wrist-spinner Kuldeep Yadav (5-99) wrapped up the innings for 300.
Australia were 6-0, still 316 runs short of making India bat again, when bad light and rain ended play early.
Only 25.2 overs were possible on a rain-hit day, meaning that the final day will start half an hour early, at 23:00 GMT on Sunday, with commentary on BBC Radio 5 live sports extra.
It was the first time in 30 years Australia had been forced to follow on in a home Test, but openers Marcus Harris and Usman Khawaja had faced only four overs before the umpires halted play just before the scheduled tea interval, and no resumption was possible.
That was frustrating for India, who lead the series 2-1 and are aiming for their first Test series victory in Australia since they began touring in 1947-48.
On a day when disgraced former captain Steve Smith and vice-captain David Warner - banned from international cricket for a year following last March's ball-tampering scandal in South Africa - were opposing skippers in a Bangladesh Premier League match, Australia's struggles with the bat were laid bare once more.
After rain wiped out the first session on Sunday, faint hopes remained when Cummins and Handscomb resumed their partnership - but fast bowler Cummins, who has shown ability with the bat in this series, was bowled by Mohammed Shami with the second new ball.
Handscomb, the last recognized batsman, added nine to his overnight score before being bowled by paceman Jasprit Bumrah, who is now the joint leading wicket-taker in the series with 21.
Nathan Lyon fell lbw to Yadav next over without scoring as Australia chose not to use a review despite having two available, leaving Mitchell Starc (29 not out) and Josh Hazlewood (21) to add an unlikely 42 for the last wicket.
But the follow-on target of 423 was still well out of Australia's grasp, and India captain Virat Kohli had no hesitation in asking them to bat again after Hazlewood became Yadav's fifth victim - his second five-wicket haul in only his sixth Test.
"It hurts, you never want to follow on in any cricket match," said Handscomb.
"It's huge for us to push for the draw tomorrow. We've got a really good chance to shift some momentum back into our camp - not just for the one-dayers coming up but for the World Cup and the Ashes - if we can last this day and show the country, show the world, that we're not far from being a really good team."