Finch completed his 16th century in the format, his innings of unbeaten 110 studded with 13 fours and two sixes.
Earlier, left-arm fast bowler Mitchell Starc, who made his ODI debut in India 10 years ago, picked up three wickets to set up Australia’s victory.
Starc struck the first blow for Australia with the new ball when he sent back Rohit Sharma for 10.
India managed to recover from that early loss through a second-wicket stand of 121 between opener Shikhar Dhawan, who top-scored for the hosts with 74, and KL Rahul.
Left-arm spinner Ashton Agar broke the stand by dismissing Rahul for 47 before pace spearhead Pat Cummins sent Dhawan back in the next over.
India needed a solid innings from captain Virat Kohli, who batted a position lower than his usual number three spot to accommodate Rahul, to get them out of trouble but he did not last long.
Kohli hit leg-spinner Adam Zampa for a six before offering a return catch to the bowler on the very next delivery to be out for 16.
Starc then returned to remove Shreyas Iyer cheaply as India lost four wickets for 30 runs to be reduced to 164-5.
Cummins and fast bowler Kane Richardson picked up two wickets each for Australia.
To compound India’s woes, wicketkeeper Rishabh Pant suffered a concussion after being hit on his helmet by a short-pitched delivery from Cummins.
The Indian cricket board said Pant, who did not come out to keep wickets, was under observation.
The two sides will meet in Rajkot for the second ODI on Friday.