India's left-arm spinner Ravindra Jadeja (5/58) picked his maiden five-wicket haul. But a record half-century by Peter Siddle (50) in the second innings took Australia to 164.
Siddle, who struck 51 in the first innings, became the first No.9 in the history of Test cricket to hit half-centuries in both the innings.
Unlike Indian spinners, Australia's slow bowlers Nathan Lyon and Glenn Maxwell failed to get the bite from the track.
Lyon, who got his career-best 7/94 earlier in the day, got enough turn but wasn't accurate.
Maxwell struck early to get rid of Murali Vijay (19), bowled through his legs while attempting a reverse sweep.
But Pujara and Kohli ran away with the match with their quick 53-run unbeaten stand for the second wicket.
While Kohli played a measured knock, Pujara was in attacking mode right from the start. Pujara struck six crisply-timed fours that left the 25,000 crowd in awe of the Saurashtra batsman.
Earlier in the day, the Australian batting crumbled on an exploding track and were bowled out for 164. The Australian tail once again wagged as Siddle (50) ended up as the top scorer for the second time in the innings.
The Indian innings lasted just 13 deliveries and they were bowled out for 272 runs in the first innings after Lyon finished with career best 7/94 and wrapped up the tail by removing Ishant Sharma and Pragyan Ojha in consecutive deliveries.
There was more drama as Australia came out to bat.
After two overs of pace from Bhuvneshwar Kumar, India skipper Mahendra Singh Dhoni was quick to unleash the spinners on the Australian batting from both the ends.
Jadeja, however, was the pick of the bowlers bagging his maiden fifer and another left-arm spinner Ojha and offie Ravichanran Ashwin bagged two each.
The demons in the Kotla track reared its ugly head as seven wickets fell, including five Australians, in 125 balls for 95 runs.
The cracks grew wider and every time a ball landed on a crack it left both the batsmen and the fielders clueless.
The Australian batsmen struggled to negotiate the Indian spinners as the ball kept low. Glenn Maxwell was promoted up the order to open the innings with David Warner.
Jadeja opened his account with Maxwell (8) and tempers flayed when he got rid of Warner (8) in his next over.
Jadeja broke into wild celebrations even before the umpire had given Warner out. Jadeja signalled Warner the way back to the pavilion.
After the send-off, umpire Aleem Dar, who was standing at square leg, cautioned Jadeja.
Dar called Tendulkar, who in return spoke to Jadeja, who was also involved in a verbal tussle with Warner Saturday during India's first innings.
Phil Hughes (6) was the next to go, trapped in front by Ashwin with a delivery that kept low.
Shane Watson (5) was also unlucky after he was castled by Ojha with a delivery that also stayed low.
Cowan became Jadeja's third victim before Steve Smith (18) and Matthew Wade (19) joined forces to repair the innings with their 53-run stand for the sixth wicket.
Jadeja struck immediately after lunch to remove Smith and Wade with successive deliveries but was denied a hat-trick by Siddle.
India pacer Ishant Sharma was also cautioned by Dar for his wild gestures after he got rid of James Pattinson (11).