India crumbled to 155 all out with just 12.5 overs left in the match and will need to win the final Test starting in Sydney on Friday to level the series and retain the Border-Gavaskar Trophy.
"What an amazing Test match," said Australia captain Pat Cummins, who was named player of the match for his six wickets and 90 runs in a pulsating contest played out in front of record crowds at the Melbourne Cricket Ground.
"I reckon that's one of the best I've been part of. I mean, all week, the crowd's been ridiculous and the cricket's been just as good."
Set an unlikely 340 to win, India looked well placed to salvage a draw when opener Yashasvi Jaiswal and Rishabh Pant batted through the second session in unflustered fashion to take their side to 112-3 at tea.
'Get the job done'
But Pant perished trying to slog part-time spinner Travis Head for six to spark a collapse as India's last seven wickets fell for 34 runs from 20.3 overs.
Top-scorer Jaiswal was out for 84 to Cummins, who took 3-28 in the innings, and Nathan Lyon ended the contest when he trapped Mohammed Siraj lbw to spark wild scenes of celebration among the home team.
Pace bowler Mitchell Starc said Australia never lost faith they could force victory in the last session.
"There were a few patches through the day with the wicket not doing a whole lot, but we managed to find a way," Starc said.
"I think that's just a feature of the group at the moment, is that calmness, to have that belief that we can get the job done."
The departure of Pant for 30 ended an 88-run partnership with Jaiswal and rejuvenated the Australia bowlers.
Ravindra Jadeja and first-innings centurion Nitish Kumar Reddy went quickly before Cummins ended Jaiswal's gritty 208-ball knock in controversial fashion.
The 23-year-old was given not out to a caught-behind appeal from an attempted hook.
But the decision was overturned by the TV umpire on visual evidence that the ball had changed direction, even though no noise was picked up by technology.
Akash Deep was caught at short leg off Scott Boland, who took 3-39, before Jasprit Bumrah scored his second duck of the match.
A tenacious Washington Sundar was left not out five after facing 45 balls.
Cummins completed an inspirational Test, with his 90 runs in the two innings being the best match tally of his career.
"Yeah, you never really know what you're going to get with my batting, so fortunately it came off this game," he said.
"We talk about it a lot, us lower order. You know, we put a lot of work into it. We know that can be the difference in a lot of Test matches."
'What a week'
Cummins removed opposing captain Rohit Sharma and KL Rahul and Starc dismissed Kohli from the last ball before lunch to leave India tottering at 33-3.
"We wanted to set the platform where the last two sessions, if we had wickets in hand, we can go for those runs. But they bowled perfectly as well," the out-of-form Rohit admitted.
"Then we wanted to fight till the end but unfortunately couldn't do it. It is pretty disappointing. We had our opportunities, we had our chances."
Rohit said Australia's final-wicket stand of 61 between Lyon and Boland in the second innings which took Australia to 234 all out was a major blow to India's victory hopes.
It came after the exceptional Bumrah, who took 5-57 in the second innings and nine wickets in the match, had reduced the hosts to 91-6 midway through day four.
The match set the record attendance for any game of cricket in Australia, with 373,591 going through the gates over five days.
"I don't think we've ever witnessed something like that so, yeah, what a week to be involved in," said Australia's Head.
"It's two bloody good teams going at it, and I felt like it ebbed and flowed throughout and it's probably one of the best Test matches that I've been involved in.
"It's just nice to come out with a win. It's been a long five days."
