Many Indian fans had a near seizure after Australia hammered their side by 10 wickets in the first 50 over game on Tuesday. But Kohli's team dominated Friday's second match and won by 36 runs, with opener Shikhar Dhawan hitting 96 and KL Rahul smashing 80 off 52 balls.
That helped to settle nerves and Kohli looked calm after the win in which he also made 78 in India's 340-6.
"We live in the days of social media and a panic button being pressed too early," Kohli said after the game in Rajkot.
"We know exactly what we're doing in the change room. On the outside there's a lot of panic that we don't really focus on."
There are worries though as Dhawan did not return to field after being hit on the rib cage by a Pat Cummins bouncer during his 90-ball knock.
Vice captain Rohit Sharma also walked off after injuring his arm diving to prevent a boundary.
"It's that left shoulder that has popped out a few times. There's no tear so hopefully he should be back for the next game," Kohli said.
India experimented with the batting order in Mumbai but stuck with two traditional openers with Kohli at three as they saved the series.