Indian spinners rattled Australia's top order after ending skipper Steve Smith's brief resistance on the opening day of the first Test in Pune on Thursday.
The visitors were 153 for four at tea with Matt Renshaw (38), who returned to the crease after earlier retiring with a stomach upset, and Mitchell Marsh (2) batting at the break.
Smith braved the heat for his dogged 95-ball 27 before falling to off-spinner Ravichandran Ashwin late in the afternoon session.
Smith was involved in crucial partnerships with Shaun Marsh (16) and then Peter Handscomb (22) before chipping a catch to Kohli at mid-wicket.
Australia lost two wickets in the space of six deliveries, with Handscomb falling to left-arm spinner Ravindra Jadeja in the previous over.
Earlier, in the first session, Australia's batting charge was halted with the key wicket of David Warner (38) and the temporary retirement of the left-handed Renshaw on 36.
The opening duo put on an impressive 82-run stand before seam bowler Umesh Yadav broke through Warner's defences before the lunch break.
Warner, who had tackled the Indian spinners with good feet movement, was bowled by Umesh off a good length delivery that hit the bat en route to stumps.
The match is the first in a four-Test series between the world's two top-ranked sides.