Mohammed Shami claimed Test-best figures of 6-56 but India struggled in pursuit of a victory target of 287 in the second Test.
It remains to be seen if Aaron Finch will be able to bat on day four of the second Test, but he was cleared of serious damage.
Australia ended day three in Perth with a lead of 175 runs over India, who collapsed after Virat Kohli's contentious exit.
While Australia face a tall order on the final day in Adelaide, Nathan Lyon still feels the hosts can take the series opener with India.
Australia appear likely to face a stiff target in the first Test against India, who ended day three 166 ahead with seven wickets in hand.
Usman Khawaja's sensational catch headlined a good start for Australia, but Cheteshwar Pujara rescued India with a fine century on day one.
Josh Hazlewood believes Nathan Lyon's performances could decide Australia's Test series against India.
Mitchell Starc and Nathan Lyon were left out of Australia's Twenty20 squad for matches against South Africa and India.
Pakistan should have the last laugh in Abu Dhabi after taking charge on a day which Azhar Ali will want to forget.
Nathan Lyon made a flying start to the second Test, but Pakistan recovered to gain a real foothold against Australia.