Cheteshwar Pujara and Virat Kohli thwarted Australia in a diligent partnership to push India to a position of strength and a 166-run lead at the close of day three of the first test on Saturday.

Brought together before tea with India vulnerable at 76 for two, the pair stood firm through the best part of a session before spinner Nathan Lyon snared Kohli for 34 to end their 71-run stand at Adelaide Oval.

Pujara, who saved India with a sparkling 123 in the first innings, survived to stumps on 40 not out, with Ajinkya Rahane on one run.

Lyon was denied Pujara's wicket twice by the Decision Review System, so his Kohli breakthrough was a relief after a long afternoon's toil.

Deceiving the Indian talisman with flight and dip, he caught Kohli's inside edge and the nick popped up to Aaron Finch at short leg.

Australia will nonetheless be desperate for early wickets on day four to have hope of restricting India to a gettable total.

Kohli was fuming to have his innings cut short but still managed to join an elite club of Indians who have notched 1,000 test runs in Australia.

Sachin Tendulkar (1,809) leads the list ahead of VVS Laxman (1,236), Rahul Dravid (1,166), Virender Sehwag (1,031) and Kohli.

Josh Hazlewood had Lokesh Rahul caught behind for 44 after fellow seamer Mitchell Starc dismissed Murali Vijay for 18 to end a 63-run opening stand, after the home side were bowled out for 235 in the rain-hit first session.

Protecting India's first innings total of 250, Kohli's bowlers did well to carve out an initial 15-run lead, getting through Australia's final three wickets for the cost of 44 runs after the home side resumed on 191 for seven.

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.

Starc was the first wicket to fall, caught behind off seamer Jasprit Bumrah, who finished with 3-47. Lyon was unbeaten on 24, having played a handy cameo with Head.

Dismissed for two in the first innings to extend a lean run of scores, Rahul came out with positive intent and smashed Pat Cummins for a six and a four with a pair of booming drives.

The recalled Vijay was never quite so settled, however, and he was out chasing a wider delivery that was caught by Peter Handscomb at second slip.

Having done the hard work, Rahul also threw away his wicket with an ill-advised swipe at a Hazlewood ball which sent a nick through to Tim Paine.

India second innings: 151/3, Pujara 40*, Rahane 1*

Fall of wickets: 1-63 (Vijay 18, 18.2 ov), 2-76 (KL Rahul 44, 24.2 ov), 3-147 (V Kohli 34, 57.1 ov)