A brilliant maiden century from Nitish Kumar Reddy powered India past the follow-on target on Saturday, after the tourists were in trouble at 5-164 in reply to Australia's 474.
Nathan Lyon (3-96) removed Reddy (114) 17 minutes into the first session on Sunday when the breakout star hit an easy catch to long-off.
It came after a controversial call from the TV umpire in the previous over when No.11 Mohammed Siraj was given not out despite the ball appearing to carry to slips.
Australia captain Pat Cummins wanted to review the decision but was unable to because TV umpire Sharfuddoula was the one who made the call.
Teenage debutant Sam Konstas will be looking to repeat his stunning Boxing Day heroics as he opens the batting again with Australia veteran Usman Khawaja.
Reddy, just 21 and playing in his fourth Test, helped India add another 11 runs to their overnight score of 9-358.
Washington Sundar and Nitish Kumar Reddy hauled India back into the fourth Test on day three. (James Ross/AAP PHOTOS)
Australia made all the early running in this fourth Test, but India turned it around on Saturday through Reddy and Washington Sundar (50) to have their best day since dominating the series-opener in Perth.
Scott Boland, the pick of Australia's bowlers with 3-57 from 27 overs, said team morale remained high.
"Obviously, it could have been better, but I think that's the Test match ebbs and flows," the 35-year-old said on Saturday night.
"Hopefully (Sunday) morning we get that final wicket, then put on a really nice lead and see how the game plays after that."
Mitchell Starc's back is causing concern as Australia strives to regain the Border-Gavaskar Trophy. (James Ross/AAP PHOTOS)
Australia bowled 120 overs at India as the effects of a gruelling series appear to take their toll on Mitchell Starc.
The star quick received treatment on his back on day three, but continued to bowl at top pace for the final session.
Starc has toiled for 25 overs in this Test, backing up from the 24 he bowled in Brisbane.
Boland was recalled to the XI after star quick Josh Hazlewood broke down for a second time in the series at the Gabba.
The fitness of Australia's bowling attack will be a major factor in deciding the outcome of the Border-Gavaskar trophy.
This Test is set to go five days, a rarity in modern cricket in Australia, meaning the players will have only three days off before the finale at the SCG gets underway.
With the five-match series tied 1-1, India will retain the Border-Gavaskar trophy if they pull off an extraordinary comeback victory in this Test.
Australia need to prevent the tourists from winning one of the last two matches of the series, while claiming at least one themselves, to regain the trophy for the first time since 2014-15.
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