The pair fell in quick succession soon after lunch to give the hosts a key breakthrough on a batting-friendly pitch in Melbourne. India lead by 435 runs.
The express paceman struck the helmet of Hanuma Vihari and then dismissed the batsman, who had been promoted to open in a new-look order, with a delivery that clipped the glove off an awkward length and sailed straight to Aaron Finch at second slip.
Pujara's record-breaking ton has taken India to a position of dominance at Melbourne and a possibility of a first-ever series win in Australia has only strengthened.
Tempers flared late in the day, with Starc fuming after Pant seemingly made a beeline for the bowler while running between the wickets.
Sharma was given a lifeline by sub fielder Peter Siddle at short fine leg as he completely botched up a pull sweep shot and Siddle had put down a sitter. Starc strikes, gets the wicket Australia needed here.
Kohli, who was stranded at 47 runs on the first day, completed his 20th Test fifty in 110 balls, while Pujara brought up his 17th Test hundred in 280 balls. While Hazlewood and Lyon got a piece each.
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Rishabh Pant (39) played an uncharacteristic defensive innings, and the only aerial stroke he played was in the 159th over off Lyon when Cummins spilled the chance at long on.
Pat Cummins was the pick of the attack, taking two for 40.
Having come together at tea on day one, Pujara and Kohli resumed at 215 for two in the morning, with the temperature already pushing 30 degrees Celsius (86F).
Pujara also scored 123 in Adelaide to help India claim the series-opener.
Australia kept a tight leash on the scoring rate as 62 runs came in this morning session.
But the exciting Agarwal rose to the challenge in front of 73,500 fans after India won an important toss and opted to bat on a pitch that is expected to deteriorate as the match wears on. The Australian captain, Tim Paine, was quite humorous, read chirpy, behind the stumps during the Boxing Day Test on Thursday.