Chasing 162 runs for a day-three victory on a tough deck, Australia wobbled early during a collapse of 3-19 before Usman Khawaja (41), Travis Head (34no) and Beau Webster (39no) put the chase to bed.
For the Australian team, a series win over India had been regarded as the final frontier, with mainstays Pat Cummins, Alex Carey, Khawaja and Head among those never to have previously held the Border-Gavaskar Trophy in their impressive careers.
Travis Head was unbeaten on 34. (Dan Himbrechts/AAP PHOTOS)
That trophy had appeared a pipe dream after a 295-run loss in the series opener in Perth, but only inclement weather in Brisbane prevented Australia from claiming wins in all four of the Test matches that followed.
But all Australia's good work threatened to come unstuck with a loss at the SCG, where defeat would have tied the series at 2-2 and kept the trophy in India's hands for a fifth consecutive series.
Scott Boland was a hero for Australia, taking 6-47 to finish with 10 wickets at the SCG. (Dan Himbrechts/AAP PHOTOS)
In reply to India's second innings of 157, Australian openers Sam Konstas (22) and Khawaja put on 39 in the first 23 balls before quick Prasidh Krishna (3-27) spearheaded India's fightback and left the hosts in jeopardy at 3-58.
Khawaja showed positive intent after lunch, punishing both Prasidh and Mohammad Siraj as India's superstar quick Jasprit Bumrah watched from the pavilion with a back issue.
But India would have felt renewed confidence when the veteran nicked to wicketkeeper Rishabh Pant from Siraj's bowling just short of his second half-century of a quiet series.
From there, allrounder Webster continued a great Test debut by forging a game-winning 58-run partnership with serial swashbuckler Head.
Beau Webster of Australia (r) and Travis Head embrace after their match-clinching partnership. (Dan Himbrechts/AAP PHOTOS)
Webster hit the winning runs down the ground a little more than an hour after the lunch break, Australia finishing at 4-162.
The win confirms Australia's spot in the World Test Championship final, to be played against South Africa at Lord's in June.
During a dramatic two-and-a-half days of cricket in Sydney, Rohit Sharma was axed as Indian captain, Sam Konstas continued to rile the tourists, India's best player Bumrah suffered a game-ending back injury and Steve Smith came up one run shy of 10,000 for his Test career.
Through it all, back-up paceman Scott Boland stood tallest, returning the best match figures of his first-class career (10-76) and his two best innings (4-31 and 6-45) since his dream Test debut at the MCG three summers ago.
Boland dispatched Siraj and Bumrah to complete his six-wicket haul and help skittle India (185, 157) within 45 minutes of Sunday's restart.
The Australian batting order (181, 4-162) sealed the result in reply to secure the 3-1 series win.
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