Australia resumed on day four at 8-179 and were dismissed for 240, setting India a target of 292 to win.
Man-of-the-match James Pattinson led Australia’s bowling, taking 4-53 in India’s second innings after also scoring 18 not out and 37 not out.
His fellow Victorian quick Peter Siddle also played a key role with figures of 3-63 and 3-42 in the match and contributing a vital knock of 41 in Australia’s first innings of 333, helping to set up a 51-run lead on the first dig.
Ben Hilfenhaus claimed 2-39 after his career-best haul of 5-75 in India’s first innings.
Sachin Tendulkar top-scored with 32 in India’s second innings.
The second Test of the four-match series starts in Sydney on January 3.
A total of 189,347 people attended the game, setting a new record for a Test match against India after beating the 181,053 who attended Steve Waugh’s final game at the SCG in January 2004.
Pattinson said the Australians had come to Melbourne under some pressure after their shock loss to New Zealand in the Hobart Test earlier this month.
“To come here and win the Boxing Day Test it’s just unbelievable,” Pattinson said.
“Victorian support, there’s nothing like it.”
India reached 1-24 at lunch before losing 5-93 in the middle session.
Virender Sehwag (7) smashed a ball to gully off the bowling of Hilfenhaus as Mike Hussey held a fine catch at 1-17.
India lost Gautam Gambhir (13) and Rahul Dravid (10) soon after lunch.
Pattinson struck when he had VVS Laxman (1) caught at square leg at 4-68 and Virat Kohli was lbw to Hilfenhaus for a first-ball duck in the next over.
Siddle raised his arms in triumph after taking the prized wicket of Sachin Tendulkar (32) caught at gully by Hussey at 6-81.
Ravichandran Ashwin (30) and Zaheer Khan (13) were both caught at short leg as Siddle and Pattinson each picked up a third wicket.
Pattinson bowled India captain MS Dhoni for 23 and spinner Nathan Lyon claimed his only wicket of the match when Umesh Yadav was caught on the boundary for 21.
Australia’s attempt to squeeze runs out of their tail began badly early on Thursday when veteran Hussey was out for 89, adding just 10 to his overnight score.
But the 10th-wicket pairing of Pattinson (37 not out) and Hilfenhaus (14) stood firm to add 43 runs in the second-highest partnership of the innings.
Australia’s ex-skipper Ricky Ponting, who made two half-centuries in the match, paid credit to the efforts of the side’s tailenders.
“The tail has done a good job for us in both innings,” he said.
“Those runs in Test match cricket are pretty crucial. It was another terrific win.”
It’s only Australia’s fourth victory in nine Tests in 2011.
Dhoni said his bowlers had done a good job to allow the tourists to face a target of 292 to win.
“It’s a decent one (total) that we should have chased,” Dhoni said.
Australia captain Michael Clarke was also thrilled with the efforts of his tailenders.
“Crucial runs in both innings. We’re calling James Pattinson Matthew Hayden the way he’s going,” Clarke said.
From Cricket Australia Website