The tourists, who were handed a victory target of 349 when Australia declared at 251-6 before the start of play, tumbled from 160-2 at tea to 217-7 midway through the session.
Ajinkya Rahane (38 not out) and Bhuvneshwar Kumar (20 not out) stopped the rot, however, and had guided India to 252-7 when Australia ran out of overs in the early evening gloom.
Despite some disciplined bowling from the Australians on a turning pitch, only one Indian batsman had fallen in each of the first two sessions and the match looked destined for a draw at the second break.
Murali Vijay had illustrated the limited ambition of the tourists when he took 222 minutes and 135 balls to score a half century but it was only when his partnership with skipper Virat Kohli was broken that an Indian victory could be discounted.
Vijay was fortunate to survive an lbw appeal off the bowling of Josh Hazlewood when he was on 46 and the paceman finally got his man when he induced the opener into an edge that Brad Haddin claimed behind the stumps for 80.
It was just reward for Hazlewood (2-31), who had bowled with great discipline and miserly economy in the first two sessions without taking a wicket.
Left-armer Mitchell Starc struck next, removing Kohli for 46 with a slanted delivery that the Indian captain edged to Shane Watson at first slip.
The tourists were now rocking at 201-4 and Starc drove home Australia's advantage in his next over by trapping Suresh Raina lbw for a duck to give the recalled batsman a pair for the match.
Off-spinner Nathan Lyon, who had removed opener Lokesh Rahul before lunch, returned to dismiss Wriddhiman Saha lbw for another duck with a delivery that took a devilish bounce off the pitch and into the batsman's pads.
The 15,000 spectators at the SCG were now buzzing and Hazlewood elicited another huge roar when he reduced the tourists to 217-7 by trapping Ravichandran Ashwin lbw for one run.
The field closed in around Rahane and Kumar with often just one Australian outside the cordon but the Indian batsmen held firm to secure a second successive draw after opening the series with defeats in Adelaide and Brisbane.
Even on a day when the Australian bowlers took centre stage, captain Steve Smith could not be kept out of the limelight and it was his brilliant diving catch that ushered Rohit Sharma back to the dressing room for 39 just after lunch.
