Debutant Beau Webster hit the winning runs as the hosts reached their target of 162 in a low-scoring match in Sydney.
Scott Boland had earlier finished with six for 45 as India – who had eked out a four-run first innings lead – were dismissed for 157.
Teenage opener Sam Konstas got Australia’s chase off to a fast start, but he skied Prasidh Krishna to Washington Sundar for 22 and Marnus Labuschagne fell to the same bowler for six.
Steve Smith followed suit for just four on his home ground, leaving him one short of 10,000 Test runs, as Australia stumbled to 58 for three.
But with the Indian attack shorn of stand-in captain Jasprit Bumrah, who took 32 wickets in the series, due to back spasms, Usman Khawaja and Travis Head (34no) pushed them towards victory.
Khawaja fell for 41, but Webster clubbed six fours as he made 39 for 34 balls to add to his first innings half century.
The win regained the Border-Gavaskar Trophy for the Australians after 10 years and sealed their place in the World Test Championship final against South Africa at Lord’s in June.