Finch (93) and Khawaja (104) laid the platform for an even bigger total in their 193-run opening stand, which spanned 191 balls and was almost double Australia’s previous highest opening partnership during David Warner’s suspension.
Kohli, fresh from his match-winning hundred in Nagpur on Tuesday, responded with a sparkling knock of 123 that came off just 95 balls.
Zampa made the crucial breakthrough, dismissing the captain for a third time on tour, and also removed MS Dhoni and Kedar Jadhav.
India were all out for 281 in 48.2 overs, with Pat Cummins and recalled paceman Jhye Richardson also playing pivotal roles with the ball.
Cummins starred with the new ball to help reduce India to 3-27 then Zampa claimed the all-important scalp of Dhoni to make it 4-86, silencing a sold-out crowd that came to see what is expected to be the wicketkeeper’s final international game in his hometown.
Kohli’s class and composure came to the fore as he wrestled momentum back from the visitors with ODI century No. 41.
The challenge for Australia was simple yet mindbogglingly difficult.
They needed to find a way to remove Kohli, whose batting average in ODI chases is now an outrageous 68.71 (with 25 tons), or lose the series.
Glenn Maxwell created a caught-behind chance when Kohli was on 98, which would have denied the superstar a ninth ODI hundred in chases of excess of 300.
Kohli, out when he skipped down the pitch and was clean bowled attempting to clip the ball to the leg side, could hardly believe the series of events.
India’s victory equation was 95 off 75 balls when Kohli departed.
Vijay Shankar and Ravindra Jadeja put together a 32-run stand but the tourists kept their cool, with the former going down swinging at Nathan Lyon before the required run-rate got out of hand.
The series continues in Chandigarh on Sunday before wrapping up in Delhi on Wednesday.
The visitors could easily have posted a far more intimidating target.
Khawaja’s mistimed pull shot triggered a collapse of 4-24 that also featured the dismissals of Maxwell, who was promoted to first drop and scored a quick-fire 47, Shaun Marsh and Peter Handscomb.