Never in their 81-year-old Test history, have India won more than two matches against Australia in a single series.
After bowling out Australia for 223 in their second innings, India chased down the target of 133 with 15 balls remaining before close of play in a match which saw a nail-biting finish.
The hosts encountered some tense moments in their run chase before skipper Mahendra Singh Dhoni smashed three fours off Mitchell Starc in a row to seal the match in India's favour much to the delight of the home crowd.
Earlier, it was a defiant 65-minute last-wicket stand between Starc and Xavier Doherty, during which they consumed 18.1 overs for 44 runs to ensure the Indians, at least, don't run away with the game before the start of the 15 mandatory overs.
When the final hour started, India required 45 runs from a minimum of 15 overs. They completed the chase losing the wickets of Murali Vijay (26), Cheteshwar Pujara (28), Virat Kohli (34) and Sachin Tendulkar (21).
Dhoni (18) and Ravindra Jadeja (8) were at the crease when the winning runs were scored.
In the process, India reclaimed the Border-Gavaskar trophy, which they relinquished after a 0-4 drubbing in Australia last year.
The victory was achieved on the back of some fine batting by debutant Shikhar Dhawan and Murali Vijay, both of them scoring big centuries in India's first innings.
Dhawan smashed 187, his 100 coming off a mere 85 balls making him the quickest-ever debutant to reach the three-figure mark.
India will now push for a clean sweep when the fourth and final final Test starts in New Delhi from Friday.