Routed for their lowest Test score of 36 in an eight-wicket loss inside three days in Adelaide before Christmas, the tourists bounced back to win by the same margin midway through day four in Melbourne.
The victory – which sets up a tantalising second half of the series between the top-ranked hosts and the third-ranked visitors – also came despite the absence of India's talismanic captain Virat Kohli, who is missing the last three Tests due to the birth of his first child.
Australia’s batting line-up has come in for heavy criticism throughout the series, having failed to pass 200 through their four innings. Batting icon Steve Smith has struggled with scores of 1, 1 not out, 0 and 8.
Having won the toss as raging favourites on Boxing Day, Australia were dismissed for 195 before India replied with 326, led by stand-in captain Ajinkya Rahane’s inspirational 112.
A second-innings collapse left Australia flirting with an innings defeat before a rescue mission led by Cameron Green’s 45 took them from 99-6 to 200 all out.
That left India needing only 70 to win, which they collected in just 15.5 overs for the loss of two wickets with man-of-the-match Rahane fittingly hitting the winning runs.
Speedster Jasprit Bumrah led the bowling for India with 4-56 and 2-54, while off-spinner Ravichandran Ashwin and paceman Mohammed Siraj each took five for the match.