Australia were bowled out for 235 before lunch on day three of the series-opening test on Saturday, leaving India with a 15-run first innings lead.
Australia resumed on 191 for seven and lost their last three wickets for the addition of 44 runs.
India seamer Mohammed Shami was on a hat-trick after claiming the final two wickets of batsman Travis Head (72) and number 11 Josh Hazlewood (0) in successive balls at the end of a rain-interrupted session.
Mitchell Starc was earlier caught behind off the bowling of paceman Jasprit Bumrah, who finished with his team's best figures of 3-47.