Pujara was the last man to be dismissed on the second last ball with Cummins coming up with a direct hit.
It was his third ton against Australia but first in Australia, and the innings only ended on 123 thanks to a sensational diving run out from Pat Cummins, which brought an end to the day's play.
Aided by a hint of bounce and movement, Australia's trio of quick bowlers did all the damage in a fiery opening session before Nathan Lyon took over after lunch. Kohli, the No.1 batsman in the world, has scored just 49 runs in his past six test innings against Australia.
Australia took four wickets before lunch with the batsmen guilty of playing at deliveries they would have been better off leaving alone.
"I thought we bowled really well for four hours, probably pretty well for another hour and probably got it a bit wrong at the end there", Starc told reporters.
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But they spectacularly rose to the challenge.
The first breakthrough came in the second over, when Josh Hazlewood had KL Rahul (2) caught in the slips by Aaron Finch, and opening partner Murali Vijay (11) joined him in the sheds when he feathered Mitchell Starc through to wicketkeeper Paine five overs later.
In marched superstar Kohli to a huge roar from the crowd, but a stunning catch by Usman Khawaja at gully off a Pat Cummins delivery saw him back in the pavilion for only three off 16 balls, with India in trouble.
The dismissal of the prolific skipper, who rates Adelaide as his favourite overseas ground, sparked wild celebrations.
But after returning from a break, Hazlewood enticed Rahane into a drive and he edged to Handscomb in the slips for 13. 25-year-old Hanuma Vihar, who made his debut at the Kennington Oval against England, was called up to the 12-member squad, although India chose to field Sharma instead.
Sharma then had a brain fade and needlessly tried the same thing on the next ball, and this time debutant Marcus Harris comfortably collected the catch, with the right-hander once more failing to capitalise on a good start.