In front of a record overall attendance for a Test in Australia of 373,691, the hosts had to get past India's defiant star opener Yashasvi Jaiswal (84 from 208), who was dramatically dismissed by Pat Cummins.
Veteran spinner Nathan Lyon (2-39), who has struggled during this series, took the final wicket with 39 minutes left and 12 overs remaining to seal an extraordinary win.
India went to tea at 3-112, seemingly set to hold on for a draw, but Cummins (3-28), Scott Boland (3-39), Lyon, and even Travis Head (1-14), Â contributed with the ball in the final session to trigger a collapse of 7-43.
Australia will regain the Border-Gavaskar trophy for the first time since 2014-15 if they win, or draw, the series decider at the SCG, starting on Friday. Victory for India and a series drawn 2-2 would allow the tourists to retain the trophy.
Pat Cummins led Australia heroically, scoring 90 runs and removing India's top three on day five. (James Ross/AAP PHOTOS)
It will be the first time a Test series in Australia heads to its finale with a trophy up for grabs since India toured here in 2003-04.
Chasing an MCG run chase record of 340, India effectively ruled themselves out of winning the fourth Test by going at a run rate of just two per over for most of their second innings on Monday.
Jaiswal, one of cricket's brightest talents, followed his 82 in the first innings with a knock that threatened to go down in Indian folklore.
The 23-year-old was fortunate to survive on 31 when the beneficiary of a contentious umpire's call.
Australia were fuming after Jaiswal was struck on his back pad and given not out, with the on-field decision remaining despite DRS showing the ball would have hit the top of leg stump.
Yashasvi Jaiswal gave India hope of survival before departing contentiously for 84. (James Ross/AAP PHOTOS)
But Jaiswal's luck ran out when he tried to hook Cummins and a not-out on-field decision was overturned on review.
Jaiswal had a heated discussion with the on-field umpires because "snicko" showed no clear noise. However, footage clearly showed a deflection of his glove.
Right from morning one on Boxing Day - when teenage debutant Sam Konstas smashed an audacious 60 that stunned India and their superstar quick Jasprit Bumrah - this was a Test that had everything.
Mitch Marsh had a lean Test but took this vital catch in the deep to remove Rishab Pant. (James Ross/AAP PHOTOS)
Bumrah yet again carried India on his shoulder with nine wickets for the match, forcing Australia into a rare tactics to contain his brilliance.
The 31-year-old, who took his 200th Test wicket at an average lower than any bowler in history, has 30 scalps for the series.
Emering allrounder Nitish Kumar Reddy scored a maiden Test century in India's first-innings fightback, but Australia had more contributors.
Steve Smith looked to have returned to his best, completing centuries in back-to-back Tests with a sublime 140 in the first innings.
Nathan Lyon took two wickets as India collapsed, including the dangerous Nitish Kumar Reddy for one. (James Ross/AAP PHOTOS)
No.3 Marnus Labuschagne, who has faced pressure to keep his spot this summer, made 72 and 70.
Cummins had a spectacular all-round match, scoring 49 and 41 to go with his six wickets.
Indian superstar Virat Kohli had a nightmare week in Melbourne; fined for a day-one shoulder bump on Konstas, involved in a dramatic day-two run-out with Jaiswal, and out for just five in the second innings.
Pressure continues to mount on Indian captain Rohit Sharma, who has made just 31 runs at an average of 6.20 this series.
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