Chasing just 133 in 36 overs for the win, time was always likely to be the biggest obstacle on day five, but they reached the target with two and a half overs to spare.
After two devastating defeats, the visitors were more competitive but a disastrous morning session when they lost five wickets for 95 runs cost them dear.
After resuming on 75 for three, Australia progressed to 170 for eight by lunch on day five, with a lead of 79, before the last two wickets put on 80 as Micky Arthur's side made 223.
Captain Michael Clarke (18) was out to Ravindra Jadeja for the fifth time in the series and Phil Hughes (69) was erroneously adjudged lbw soon after.
While Australia certainly suffered some bad luck, the lack of fight in the lower-middle order was alarming and the news that vice-captain Shane Watson was on his way back to India, having recommitted himself to the side, was surely welcome.
Nightwatchman Nathan Lyon was the first man to go, caught behind off Pragyan Ojha, although he could have been caught at slip two overs earlier.
Clarke, who had been unable to bat at his new elevated position of three the previous night because of back pain, joined Hughes at the crease and quickly helped himself to a couple of boundaries off Ojha.
But it was a rare attacking highlight for the tourists as the pair set themselves to bat out time.
With Clarke apparently affected by his back problem, though, he never looked comfortable and he fell to Jadeja for the fifth time in six innings as he got an inside edge on to his pad and was caught by Cheteshwar Pujara at short leg.
Three overs later the normally reliable Aleem Dar gave Hughes, who had added 16 to his overnight 53, out lbw to a ball from Ravichandran Ashwin which was clearly missing leg stump.
When Moises Henriques (two) and Peter Siddle (13) failed to stick around long and with Australia 143 for eight - 52 ahead - there was a possibility the match could be over before lunch.
But Brad Haddin and Mitchell Starc, who scored 99 in the first innings, at least saw their side through to the interval.
After a 36-run stand Australia were nine down when Haddin was trapped plumb in front off Ashwin for 30 to leave them 179 for nine.
Starc and Xavier Doherty dug in and added 44 for the final wicket before the former was finally dismissed for 35, caught by Ashwin off the bowling of Jadeja.
India lost their first wicket with their score on 41 when Murali Vijay advanced down the pitch only to miss Doherty's delivery, with Haddin whipping the bails off.
India looked to be coasting to victory at 70 for two, but their momentum was temporarily halted when Pujara went for 28 after he played around a Lyon straight one and was given out lbw.
Time was the only barrier between the home side and victory, although 45 runs in 15 overs should not have proved too big a task for Mahendra Singh Dhoni's side.
Virat Kohli and Sachin Tendulkar looked to be leading their side to victory before the former went for 34, chipping to Hughes at midwicket off the bowling of Siddle to leave them 103 for three.
Bizarrely Dhoni made six from 24 before Tendulkar decided enough was enough, smashing Starc back over his head for four.
There were still to be nervy moments for the home side and Tendulkar (21) was run out after he was turned back for a single by Dhoni, leaving the hosts needing 17 from 23.
Jadeja is not one to die wondering and his mis-timed lofted drive off Siddle trickled to the fence before another straight drive two deliveries later yielded four more.
Dhoni joined in with a lofted drive through the covers and followed it up with a flick off the pads which brought the scores level.
The skipper ended the game in style the next ball with a pull to the boundary as India confirmed their dominance over the tourists.