Chasing 323, the hosts were bowled out for 291 in 119.5 overs shortly before tea on day five with Ravichandran Ashwin, Jasprit Bumrah and Mohammed Shami taking three wickets each for a 1-0 lead in the four-match series.
India gave wings to their ferocious ambition of winning a maiden series Down Under, beating Australia by 31 runs in the opening Test with an irresistible fusion of self-belief, hunger and talent on Monday.
Things went too close for comfort for India as Nathan Lyon (38 not out) and Josh Hazlewood (13) put on 42 runs for the last wicket and frustrated the Indian bowling. The sparse crowd at Adelaide Oval cheered every single as the duo edged closer and the odd boundary didn't help matters.
Post lunch, India had a big early breakthrough when Australian skipper Tim Paine (41) played an uncharacteristic pull shot off Bumrah and only managed to loop it up for Pant to take an easy catch.
Australia replied with 235 and conceded a 15-run lead. The visitors then finished at 307 in their second innings, including a collapse of five for 25, and set a competitive target on day four.
"It's important to stay calm. The odds were stacked up against them as soon as we got Pat Cummins out. I wouldn't say I was cool as ice but you try not to show it," Kohli said at the end of the match.