India, who won the first two matches, lost three early wickets but Kohli smashed a magnificent 123 to briefly rekindle local hopes before the hosts were bowled out for 281 in 48.2 overs.
Earlier, Finch walked out hoping to end a dip in form having gone without a fifty in his last nine ODI innings after counterpart Kohli won the toss and elected to field.
Finch clobbered Kedar Jadhav for a six and followed it with a boundary to bring up his fifty and celebrated by sending the next delivery from the part-time spinner over the ropes for the second of his three sixes.
The century, however, did not materialise and Kuldeep Yadav (3-64) trapped him leg before.
Khawaja, spilled at backward point by Shikhar Dhawan early in his innings, could not be denied his maiden ODI century.
Having raised 193 with Finch, Khawaja took a single off Ravindra Jadeja to bring up his century, taking his helmet off and looking upwards in a muted celebration.
Mohammed Shami eventually dismissed Khawaja whose 104 will boost his claims for a place in Australia's World Cup squad.
A promoted Glenn Maxwell smashed three sixes in his busy 47 before Australia, cruising at 244 for two in 40 overs, suffered a mini-collapse even though the 300-mark was never in doubt.
Pat Cummins and Jhye Richardson rocked India's chase early, reducing them to 27-3 in the seventh over leaving Kohli and Mahendra Singh Dhoni with a mountain to climb.
Dhoni departed for 26 in what many feel could be the 37-year-old's last knock in his home city but Kohli never looked in trouble.
The only blemish in Kohli's knock was when Alex Carey dropped him with the batsman two short of his second successive century and 41st in ODIs.
Zampa pegged back Kohli's leg stump to cut short a stellar knock and India's lower order failed to get them over the line.
India will carry a 2-1 lead into Sunday's fourth match in Mohali.
Kohli and his team mates wore army camouflage-style caps in solidarity with Indian paramilitary police killed in a militant attack by a Pakistan-based group in an unusually strong display of patriotic fervour in sport.
(Reporting by Amlan Chakraborty in New Delhi; editing by Toby Davis)
