Australia was all out early into the final day with a lead of only 49 and first time sensation Moises Henriques unbeaten on 81 following his first innings 68.
India won comfortably by eight wickets.
The home nation took just 11.3 overs to wrap-up victory and conclude a barnstorming victory at 2-50, despite openers Verendar Sehwag (19) and Murali Vijay (six) falling as closing stage casualties.
Batting maestro Sachin Tendulkar was unbeaten on 13 and Cheteshwar Pujara eight to complete Australia's last rites 45-minutes before lunch on the fifth day.
Tendulkar belted 12 with successive sixes from the first two deliveries he faced off Nathan Lyon as world cricket's most decorated batsman, in his 196th Test, arrived at the crease and his side needing just 14 to win.
Big Aussie quick James Pattinson and a much maligned Lyon grabbed a morale-boosting wicket each.
Australia was all out for 241 when number 11 Lyon popped a simple catch into short-leg from left-hand spinner Ravi Jadeja just 24 minutes into the last day.
Lyon bravely held out for 77 balls in support of Henriques as the last pair ironically put on 66 in the biggest partnership of Australia's innings.
Henriques survived 148 deliveries in his impressive innings which included six fours and two big sixes.
Australia resumed the last day at 9-232 and a slender lead of only 40 runs, with forlorn hopes of victory against a rampant home nation on a terribly worn and dusty Chennai pitch.
Henriques was completing his fine debut double at 75 and Lyon was on eight.
The last pair defiantly put on 57 late on the fourth day when it had seemed that Australia could have been dealt a demoralising innings defeat.
The confident home nation put together an imposing 572 in their first innings on the back of a stunning career-best 224 from captain Mahendra Singh Dhoni and 107 from Virat Kohli, in reply to the Aussies first-up 380.
Dangerous off-spinner Ravi Ashwin mesmerised the Aussies with 12 wickets for the match from his 7-103 from 42 first-innings overs and then 5-95 from another 32 overs in the second.
Ashwin with stunning match figures of 12-198 was main man-of-the-match candidate with his skipper, Dhoni, who won the coveted award for his fifth Test centruy and first against Australia.