Australia resumed on 274 for five at the Gabba, having finished day one in a promising position courtesy of Marnus Labuschagne's 108, his fifth test century.
Their hopes of building a big total were thwarted, however, as India's bowlers rallied superbly to remove captain Tim Paine (50), Cameron Green (47) and Pat Cummins (2) for the loss of only four runs midway through the session.
Nathan Lyon soaked up his 100th test and the acclaim of the Gabba crowd as the spinner pushed Australia past 350 before he was bowled by debutant Washington Sundar for 24 off 22 balls.
Mitchell Starc finished unbeaten on 20, after paceman T. Natarajan, the second of India's test debutants, wrapped up the innings by bowling Josh Hazlewood for 11.
Natarajan finished with 3-78, while Sundar and Thakur also grabbed three wickets apiece.
Paine was caught in the slips by Rohit Sharma off Thakur, with Green following moments later after being bowled by Sundar.
Thakur trapped Cummins lbw, with the tailender failing to overturn the decision on review.
India, who lost the toss and were sent in to bowl on day one, are shorthanded in attack, with seamer Navdeep Saini having suffered a groin injury.