After India axed their captain Rohit Sharma and slumped to 2-17 in the first hour, Kohli edged the first ball he faced to the slips cordon.
Steve Smith grabbed the ball and scooped it up to Marnus Labuschagne in the gully, only for third-umpire Joel Wilson to decide the ball had "rolled" on the ground.
Kohli then went to lunch unbeaten on 12, as Shubman Gill was caught at slip off Nathan Lyon for 20 on the last ball of the first session at a sold-out SCG.
After claiming Rohit had "opted out" of the match, India won the toss and batted in Sydney but immediately had three balls beat the bat in the match's second over.
KL Rahul clipped Mitchell Starc straight to Sam Konstas at square leg on four, before Scott Boland angled one across Yashavi Jaiswal and drew his edge on 10.
India then looked at risk of slipping to 3-17 when Kohli edged Boland in his first ball, before being handed the reprieve.
Kohli and Gill put on 40 for the third wicket, before the latter jumped down the pitch to Lyon and edged him to Smith at slip on the last ball of the session.
The tourists must win the final Test in Sydney in order to retain the Border-Gavaskar Trophy, with a draw at the SCG enough for Australia to clinch a 2-1 series victory.
Kohli looms as their biggest hope.
While his second-innings century in the opening Test in Perth has been his only score above 40, Kohli has long stood up on big occasions for India.
After putting his cover-drive away for the most part in Melbourne, the right-hander's first two scoring shots on Friday came to balls outside off stump.
Debutant Beau Webster (3-l) was in the action early by taking the catch to dismiss Yashasvi Jaiswal. (Dan Himbrechts/AAP PHOTOS)
He has been watchful since, playing 14 straight dot balls just before the break with Australia debutant Beau Webster bowling a stump-to-stump line.
A series win over India would hand Australia back the Border-Gavaskar Trophy, and mean Pat Cummins' side hold every bilateral Test series trophy available to them.
A win at the SCG would also be enough to ensure qualification for the World Test Championship final against South Africa at Lord's in June.
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