The Indians took about 25 minutes in the morning to finally bring an end to the stubborn last wicket partnership between Moises Hendriques (81 not out) and Nathan Lyon (11) as Australia folded up for 241 in their second innings, leaving the hosts a small victory target of 50.
The hosts lost both their openers Murali Vijay (6) and Virender Sehwag (19) before overhauling the target with more than two full sessions to spare.
Cheteshwar Pujara brought about the winning run off Nathan Lyon to trigger off scenes of celebrations in the galleries and the Indian dressing room as the players congratulated and hugged each other.
It was a commendable show by the Indians, who have been under pressure of late following a series of indifferent performances in recent times, as they dominated the game right through at the M A Chidambaram stadium where the conditions suited the home team.
While captain Mahendra Singh Dhoni expectedly got the man-of-the-match award for his game-changing knock of 224, off spinner Ravichandran Ashwin scalped 12 wickets in the match to also play a stellar role.
The margin of victory could have been bigger had the Indians managed to get rid of the last wicket pair yesterday itself and had they not lost both their openers despite chasing a small total.
The two teams will now travel to Hyderabad for the second Test be played at the Rajiv Gandhi International stadium from March 2 to 6.
This was the third time that Indian spinners managed to take all the 20 wickets in a match.
The first time that Indian spinners bagged all the 20 wickets was at this same venue against England in 1972-73 while the second such instance was against New Zealand in Auckland in 1975-76.
Resuming at the overnight score of 232 for nine, Australia could add just nine runs to their tally much to the relief of the Indians who had a frustrating time trying to dismiss the last-wicket pair.
It took the Indians just nine overs this morning to put an end to Australia's resistence when Ravindra Jadeja broke the agonisingly long last-wicket partnership between Henriques and Lyon.
Lyon, who showed a lot of grit and temparament to hang on, was dismissed by left-arm spinner Jadeja, caught by Murali Vijay at short-leg.
Henriques followed up his first innings half century with another gutsy unbeaten 81 and together with Lyon added 66 runs for the final wicket to take the match into the fifth and final day and prevent India from recording an innings victory.
Chasing the paltry 50-run target, India suffered two early blows in the form of openers Sehwag and Vijay, whose flop show with the bat continued.
The opening duo of Sehwag and Vijay put on 16 runs before the former was the first one to depart.
Vijay started his innings with a bang, smacking Lyon straight over his head for a six, his only scoring shot of the innings.
But that was all the local boy could manage in the innings as he was caught by Henriques at mid-off off James Patinson when he tried to go a big drive in the next over.
Australia 1st Innings: 380 all out
India 1st Innings: 572 all out
Australia 2nd innings (Overnight 232 for 9):
Moises Henriques not out 81
Nathan Lyon c Vijay b Jadeja 11
Extras: (B-15, LB-2) 17
Total: (all out in 93 overs) 241
Fall of wickets: 1-34, 2-64, 3-65, 4-101, 5-121, 6-131, 7-137, 8-161, 9-175, 10-241
Bowling: Ashwin 32-6-95-5, Harbhajan 27-6-55-2, Jadeja 31-8-72-3, Ishant 3-1-2-0.
India 2nd innings:
Murali Vijay c Henriques b Pattinson 6
Virender Sehwag c Clarke b Lyon 19
Chesteshwar Pujara not out 8
Sachin Tendulkar not out 13
Extras: (B-4) 4
Total: (for 2 wickets in 11.3 overs) 50
Fall of wickets: 1-16, 2-36
Bowling: Pattinson 3-1-13-1, Lyon 5.3-0-29-1, Siddle 3-2-4-0.