VIDEO: India v Australia Day 5 highlights
According to the supersport.com website, Mahendra Singh Dhoni hit the winning runs via three consecutive boundaries, but the hosts made heavy weather of the chase in the final hour, needing 45 runs off 15 overs.
Peter Siddle induced a mistimed pull from Virat Kohli that Phil Hughes bagged at mid-off. Suddenly the runs dried up, and between Siddle and Mitchell Starc, they kept India quiet for 24 deliveries which only yielded five runs.
Sachin Tendulkar broke free with a lofted boundary over mid-off as his team needed 22 runs off 24 balls, but was run out going for a leg-bye.
India needed 17 off 17 when Ravindra Jadeja strode to the crease and promptly dispatched Siddle for two boundaries to release the pressure.
The shackles were broken and Dhoni finished it off the next over.
Earlier, India were 30 without loss at tea after they dismissed Australia for 223, chasing a target of 133 for an unassailable 3-0 lead in the four-match series.
Australia's total was boosted by a stubborn 44-run partnership between Starc and Xavier Doherty before the wicket of Starc - Australia's batting hero in the first outing - brought the innings to a close.
Starc faced 100 deliveries for his 35 runs, giving Ravindra Jadeja his third scalp. He conceded only 35 runs in 16.2 overs.
Doherty finished unbeaten on 18. The final pair frustrated the hosts for more than 18 overs after losing Brad Haddin for 30 post-lunch.
Bhuvneshwar Kumar also bagged three wickets for 31.
India's spinners ripped through Australia's middle order in the morning session, as the visitors lost 5-54 before Haddin and Starc consolidated, reaching lunch on 170 for eight, a lead of only 79 runs.
The visitors resumed the day on 75 for three, but they lost night-watchman Nathan Lyon in the first 30 minutes of play, still two runs adrift of India's first innings total.
Michael Clarke (19) fell victim to Jadeja for the fifth time in the series pushing a full delivery onto his pad for Pujara to take the catch at short leg.
Within seven runs, overnight batsman Phil Hughes (69) as well as Moises Henriques found themselves back in the hut as well, courtesy of Ravichandran Ashwin and a blinding caught-and-bowled effort from Jadeja.
Siddle, having scored a run-a-ball 13, was clean bowled by Pragyan Ojha.
The teams now move to Delhi for the final encounter, starting on Friday, March 22.