Finch, who replaced young Hilton Cartright after recovering from a calf injury, made an instant impact with a classy 124 after skipper Steve Smith won the toss for the first time in the series and elected to bat.
Trailing 0-2 in the five-match series, the Australians desperately needed a top-order batsman to produce such a knock.
Exploiting a flat Holkar track, Finch added 70 runs for the opening wicket with David Warner (42) and raised a 154-run partnership with Smith for the second wicket.
Finch, whose knock came off 125 balls, decorated his eighth ODI century with 17 boundaries including five sixes.
A docile pitch to deal with, Indian bowlers hardly troubled the Australians as Kuldeep Yadav (2/75) and Yuzvendra Chahal (1/54) largely remained ineffective.
Finch's dominance against Kuldeep was clear from the fact that he collected 41 runs off the 26 balls he faced from the Chinaman.
However, the hosts staged a remarkable comeback in the last 10 overs of the game after the fall of Finch, giving away just 59 runs and taking four wickets.
The stage was set for Australia to go for the kill, being placed 234 for two in 40 overs, but they lost Smith and Glenn Maxwell (5) in successive balls.
Kuldeep saw the back of Smith, caught at long-off, while Chahal had Maxwell stumped.
Bumrah got rid of Travis Head (4) and Peter Handscomb (3), leaving Australia in tatters. They could never get the fillip, which was there to take after a stupendous start even as Marcus Stoinis took the side close to 300 with his unbeaten 27.
There was hardly any help for the bowlers but both Bhuvneshwar Kumar and Bumrah remained disciplined.
The visitors scored 49 runs in the first 10 overs with Warner finding three boundaries off Bhuvneshwar and Finch hitting the same number of fours off Bumrah in this period.
Captain Virat Kohli replaced them with Chahal and Hardik Pandya. Warner, who had got his eye in, hit the first six of the innings as he lofted Chahal flat and hard over long-on.
Pandya provided India their first breakthrough by dismissing Warner, who did not move his feet while trying to punch it away and was bowled.
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