After being sent in to bat by England captain Eoin Morgan, the home side posted 304 for eight from their 50 overs.
Finch's century was backed by fifties from Mitchell Marsh and Marcus Stoinis as Australia recovered from a shaky start in which the tourists finally managed to get bogeyman Steve Smith cheaply.
Fresh from scoring three centuries as Australia won the Ashes 4-0, Smith looked in commanding touch again as he scored at better than a run a ball, but was caught behind from the bowling of spinner Adil Rashid for just 23.
But burly opener Finch filled the void with a dashing hundred, his ninth in one-day internationals and fourth against England.
Finch set the tone when he hit two of the first three balls he faced for four and struck the ball powerfully in making 107, smashing 10 fours and three sixes.
He brought up his century from the 112th ball he faced with his third six, a huge hit over long on from the bowling of Moeen Ali.
However, Ali got his revenge shortly afterwards, with Finch holing out to Jonny Bairstow at deep mid-wicket.
Marsh continued his renaissance from the Test series with 50 from 66 balls, but was bowled by a cleverly disguised wrong'un from Rashid.
He put on 118 with Finch for the fourth wicket after Australia were in a little trouble at 78 for three following the dismissal of Travis Head for five.
Stoinis enjoyed his return to the Australian team with a dashing 60 off just 40 balls, including five fours and two sixes.
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