Ravindra Jadeja claimed a career-best 7-42 as he and Ravichandran Ashwin (3-59) routed Australia, engineering a spectacular batting collapse that left India needing 115 runs to go 2-0 up in the four-test series.
India's top order did wobble but the meagre target meant their victory, which came in the second session on day three, was never in doubt.
Skipper Rohit Sharma made 31 and Cheteshwar Pujara, who made 31 not out, scored the winning run in his 100th test.
"It's been a great test match," Pujara said after guiding India to 118-4. "It is really special feeling hitting the last boundary. We are still looking to win the next two test matches."
India, who also had triumphed inside three days in the opening test in Nagpur, too, are now on the verge of qualifying for the World Test Championship final in June.
Jadeja was adjudged player-of-the-match for his match haul of 10 wickets.
"Fantastic result for us," Rohit said. "Looking at how things were yesterday. How we came back and finished the job was a great effort, particularly from the bowlers."
Earlier, resuming on a promising 61-1, Australia added 52 runs before Jadeja and Ashwin engineered a spectacular collapse.
See the scorecard of the second test at Flashscore
AUSTRALIA COLLAPSE
Asked to open in place of a concussed David Warner, Travis Head drove the third ball of the day from Ashwin for a boundary to signal his attacking intent.
The off-spinner, however, had his revenge three balls later when he drew Head (43) forward to induce an edge that Srikar Bharat collected behind the stumps.
Once Jadeja sent back Marnus Labuschagne (35), the wheels came off Australia's innings with nine of the 11 batsmen posting single digit scores.
On a pitch where spinners ruled the roost, several Australian batsmen fluffed their sweep shots to be either bowled or lbw.
Their collapse on Sunday was reminiscent of their second innings meltdown in Nagpur where they were all out for 91 losing all 10 wickets in one session.
Australia had made a bold selection gamble picking a three-pronged spin attack and going with skipper Pat Cummins as the sole fast bowler.
Opener David Warner suffered concussion, and a fractured arm, and Matt Renshaw replaced him for the match.
"I thought 260 was a decent score on that first day wicket (but India) batted quite well," Australia skipper Pat Cummins said, referring to their first-innings 263.
"I think this one is particularly disappointing, having been ahead of the game for a lot of it. Those opportunities don't come often in India, so you've got to grab them.
"So unfortunately we missed this one so, yeah, this one hurts a bit.
"The review over the next few days will be what we could have done differently."
While Australia continue to lead the standings, India would seal a place in the WTC final with victory in the third test in Indore beginning on March 1st.
