The hosts resumed on 248-6 and, helped by Ravindra Jadeja's 63, reached 332 - a first innings lead of 32.
Jadeja, fellow spinner Ravichandran Ashwin and pace bowler Umesh Yadav took three wickets apiece as Australia were bowled out for 137 in 53.5 overs.
India reached 19-0 at the close after KL Rahul and Murali Vijay negotiated the final six overs of the day.
Jadeja's sparkling knock with the bat, which included four sixes, dominated a 96-run seventh-wicket partnership with wicketkeeper Wriddhiman Saha that saw India move past Australia's first innings total.
In their second knock, Australia were 31-3 and only Glen Maxwell offered any real resistance with 45 as the tourists lost their last seven wickets for 50 runs.
The four-Test series is tied at 1-1.
India need 87 runs to win the series-deciding fourth Test against Australia after dominating the third day.