The hosts were bowled out for 200 early on the fourth day, leaving India needing only 70 to win at the MCG.
Debutant Mohammed Siraj took 3-37, including Cameron Green, who offered some resistance in making 45 for Australia.
India slipped to 19-2 before Shubman Gill, another debutant, and stand-in captain Ajinkya Rahane saw them home.
Gill, who made 45 in the first innings, finished unbeaten on 35 and man of the match Rahane followed up his first-innings century with 27 not out.
It was a remarkable turnaround for India, who were bowled out for 36 in an eight-wicket defeat in Adelaide just over a week ago.
The third Test of a thrilling four-match series begins on 6 January in Sydney.
‘One of the great comebacks’ – reaction
India head coach Ravi Shastri: “To be rolled over for 36, and then three days later, to get up and be ready to punch was outstanding.
“This will go down in the annals of Indian cricket, world cricket, as one of the great comebacks in the history of the game.”
India legend Sunil Gavaskar, speaking to Alison Mitchell: “The most pleasing thing was the determination of the entire team. The way they carried themselves – there was nothing negative from day one.”
How did India turn it around?
Just 10 days ago India were humiliated at the Adelaide Oval as they were bowled out for their lowest total in Test cricket.
They were not at full strength in Melbourne: captain Virat Kohli has returned home for the birth of his first child, fast bowler Mohammed Shami is out of the tour with a fractured arm and Umesh Yadav missed much of the match after injuring a calf.
However, man of the match Rahane set up victory with a superb century – made over six hours – in the first innings. He now has three wins from three Tests as captain.
He was supported by opener Gill, who overcame a fearsome Pat Cummins spell with the new ball, and Ravindra Jadeja later took up the mantle with 57 from number seven.
With Shami ruled out and Umesh unable to field, 26-year-old Siraj impressed as Australia – trailing by 131 on first innings – added 67 to their overnight 133-6 in the second.
Siraj was not certain to play in Australia. His father died in November, shortly after India had arrived in Australia, but, after speaking to his mother and Kohli, he opted to stay on the tour.
With India awaiting news of Yadav’s fitness, Siraj could be key as they aim to complete what would be a remarkable series win.
Problems for Australia?
Australia will have to improve on what captain Tim Paine described as “poor, sloppy cricket”.
No batsman made 50 for the hosts – the first time an Australia team has failed to do so in a home Test since 1988-89 – and they have been dismissed in this series for 191, 195 and 200, before chasing down 93 for victory in the first Test.
Opener Joe Burns has made three single-figure scorers in four innings, while Steve Smith – the number one Test batsman in the world – has a highest score of eight this series.
Australia also have injury problems of their own, with opener David Warner missed the first two Tests because of a groin issue.
“David looks really good from what I’ve seen. He’s been training this week in the nets and has started running between the wickets,” Paine said.
“The early signs with him are very good for the third Test, which is awesome for us.”
Australia were later fined 40% of their match fee and docked four ICC World Test Championship points for a slow over-rate.
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