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.
Nightwatchman Nathan Lyon was the first man to go, caught behind off Pragyan Ojha. Captain Micahel Clarke, who had been unable to bat at his new elevated position of three the previous night because of back pain, joined Phil Hughes at the crease.
Clarke never looked comfortable and he fell to Ravindra 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 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, saw their side through to the interval.
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's momentum was temporarily halted when Pujara went for 28 after he played around a Lyon straight one and was given out lbw.
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.
Tendulkar (21) was run out after he was turned back for a single by Mahendra Singh Dhoni, leaving the hosts needing 17 from 23. The skipper ended the game in style the next ball with a pull to the boundary as India confirmed their dominance over the tourists.
It is the first time Australia have lost the first three Tests of a series since 1988-89 against West Inidies.
The fourth and final test comes in Delhi between March 22 and 26.