Travis Head and Marnus Labuschagne held their nerve on a viciously turning Indore pitch to guide Australia to a nine-wicket victory Friday and only their second Test win in India since 2004.
After Australia were thrashed in the first two Tests inside three days, the series now stands at 2-1 to the hosts with one match remaining.
Labuschagne finished on 28 not out and Head on 49, having lost opener Usman Khawaja to only the second ball of the day.
The dogged victory secured Australia a berth in the ICC World Test Championship final in June at The Oval. India will be sure of their place in the final if they win the fourth Test in Ahmedabad.
"I just tried to take it one step a time," said Head.
"We have seen it throughout the series that with the wickets, the quality of bowling, anything can happen.
"So we were just trying to take it one ball at a time. It's just nice to contribute."
In a low-scoring and frenetic encounter, Australia skittled India for 109 on day one with spinner Matthew Kuhnemann giving the hosts a taste of their own medicine with five wickets.
In reply Australia made a solid start before collapsing to 197 all out before lunch on day two, their last six wickets tumbling for just 11 runs.
Eight wickets for spinner Nathan Lyon saw India bundled out for 163 in their second innings, setting up a victory target of 76.
Australia made the worst of starts with Khawaja, the hero of the first innings, out second ball of the day.
The opening batsman edged Ravichandran Ashwin to keeper Srikar Bharat for a duck.
A stunned Khawaja, who had looked supremely assured in the first innings, reviewed the decision but the dismissal was upheld -- to jubilation from the noisy Indore crowd.
Virat Kohli was convinced he had caught gum-chewing Labuschagne, the world number one Test batsman, at slip off Ashwin in the seventh over but India's review was unsuccessful.
After 45 minutes, Labuschagne and Head changed gears.
Head launched Ashwin for a six over mid-on and the next over drilled Ravindra Jadeja down the ground for a four as the shackles came off.
Labuschagne too got in the act, hitting Jadeja for a four and dismissing a short Ashwin delivery on a half-volley for four more to bring a rare Australia win in India in view.
A flurry of more boundaries quickly reduced the deficit, calmed nerves in the Australian dressing room and silenced the home crowd.
It is only the third time that India -- who have won their last 15 home series - have lost a game at home in the last decade.