Finch (121) and fellow opener David Warner (65) dazzled while captain Michael Clarke contributed 43 brisk runs as the hosts eased to a victory target of 270 with 4.2 overs to spare.
Earlier, half-centuries from Gary Ballance (79) and Eoin Morgan (50) helped England put behind a wobbly start and post 269 for seven wickets after captain Alastair Cook had opted to bat first.
The 163-run stand between Finch and Warner in 28 overs laid the foundation for Australia's comprehensive win which prolonged England's misery Down Under after their Test series whitewash.
Australia's openers enjoyed their share of luck with Finch dropped by Ballance on eight and Warner, when on 22, called back from the boundary line after replays could not establish conclusively Jos Buttler had taken a clean catch behind the stumps.
Warner eventually holed out in the deep off Joe Root to depart after a 72-ball knock that included five boundaries and a six off Chris Jordan.
Jordan dismissed Shane Watson for a two-ball duck in the next over but Clarke and Finch scored freely to kill the contest.
Balance eventually caught Finch in the 40th over off Ben Stokes, ending his 128-ball blitz.
England suffered an early blow when Cook edged the fourth ball of the day to perish caught behind for four.
Clint McKay (3-44) claimed his second victim in Root (three) to reduce the tourists to 22 for two before Ian Bell and Ballance steadied the ship.
Bell reverse-swept Xavier Doherty for a boundary and slog-swept Glen Maxwell for a six to reach 41 before attempting a repeat of the same shot only to be clean bowled by Doherty.
Morgan joined Ballance in the middle for England's most productive partnership, adding 83 runs from 14.1 overs with some cheeky shots and sharp running between the wickets.
Maxwell dismissed Morgan for 50 off 47 balls while Ballance fell in the 46th over, having justified his call-up with his maiden ODI fifty.
Man of the match Finch said: "It's nice, it's always good to get a good start to the series, it's a nice feeling.
"I think the way the wicket was, it swung around a bit early. You need a bit of luck in one-day cricket. It was nice to go on and get a big score and contribute in a good partnership with Davy Warner.
"Sometimes it works, sometimes it doesn't. He(Warner) takes a lot of pressure when you bat with him, I've known him for the best part of 15 years."
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- David Warner
- Gary Ballance
- Eoin Morgan
- Alastair Cook