A rattled India limped to 51-4 at stumps on a day when the Australians were only able to bowl 17 overs at the visitors and players left the field seven times due to downpours.
KL Rahul was on 33 at the end of a wet day along with skipper Rohit Sharma, who was yet to score.
Needing 245 to avoid the follow-on, India lost three early wickets in the first session, after earlier bowling Australia out for 445.
Yashasvi Jaiswal, Shubman Gill and Virat Kohli all fell to Australia's opening bowlers Mitchell Starc and Josh Hazlewood.
They then lost Rishabh Pant just before tea, which was called early because of another downpour that then kept the players off the field for over two hours.
When they did return, bad light saw the Australians throw the ball to off-spinner Nathan Lyon and part-timer Travis Head.
But after only 2.5 overs, umpires Richard Kettleborough and Richard Illingworth decided that the light was not good enough and called an end to play.
With most of the first day also lost to rain Saturday, Head and Steve Smith's second-day centuries ensured that India cannot realistically win the Test.
More rain is forecast for Tuesday and Wednesday, raising India's hopes of salvaging a draw and heading to Melbourne with the series locked at 1-1.
However, Australian batsman Mitchell Marsh stressed there was enough time for a result. "We know we have to take 20 wickets to try and win this Test match," he said.
"I guess all the conversations and all the planning will be how do we do that.
"There's two days to go so there's still plenty of time to get a result, so let's hope the weather stays away over the next two days."
Australia resumed the third day on 405-7 and were eventually bowled out for 445, wicketkeeper Alex Carey the last man out after an enterprising 70.
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