Finch and Warner put on 187 in 29.3 overs for the first wicket to lay the foundation for Australia's total and continue the hosts' overwhelming dominance of the series it leads 3-0.
The pattern of that domination changed on Wednesday; India batted first in each of the first three matches and lost.
Finch reached his seventh one-day international century from 97 balls. It was his first against India and his second at Manuka Oval in the Australian capital.
Warner had not developed any rust during his absence from Australia's team following the birth of his second daughter.
He was the main aggressor in a blazing beginning by the Australian openers who rushed to 40-0 after five overs. Warner posted his half century from 46 balls with a six off Gurkeerat Singh, which followed eight fours.
The pair slowed a little before Finch achieved his half century from 61 balls. He became the more aggressive of the two and set the tempo of the innings.
"Finchy towards the end there said to me to just keep batting normally and he'd up the ante," Warner said. "The plan was to get to 200 first and then go from there."
Warner fell in the 30th over. A ball from Ishant Sharma slanted across him and, in attempting to play a forcing shot to off, chopped the ball onto his middle stump.
Mitchell Marsh was promoted above captain Steve Smith to first drop but scored only two runs from his first 15 balls.
Finch was out when Australia was 221-2 in the 38th over after he and Marsh had added only 34 runs in 8.4 overs. The opener reached 100 with a single from his 97th delivery, two years after his first century at Manuka Oval against South Africa.
Finch was finally caught off Umesh Yadav, skying a pull shot to Ishant Sharma at mid-on.
Finch's power-hitting claimed one notable victim. In the 17th over, his straight drive hit English umpire Richard Kettleborough on the right leg and Kettleborough was forced to retire from the match. He was replaced by Australia's Paul Wilson.
Smith hit the ball powerfully from the outset and reached his 50 from 27 balls with four fours and three sixes. Australia was just short of 300 when Smith was out in the 46th over, caught by Gurkeerat Singh off Ishant Sharma.
It then lost 5-37 to lose momentum before Glenn Maxwell hit 41 from 20 balls with six fours and a six, falling to the last ball of the innings.
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