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.
Dangerman David Warner lasted just five balls -- and one of those was a dropped catch -- in making two.
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.
Finch survived a third umpire review on an LBW appeal from the bowling of Rashid, and Ali also dropped a tough caught-and-bowled chance when the right-hander was at 61.
Marsh continued his renaissance from the Test series with 50 from 66 balls, but was bowled by a cleverly disguised wrong'un from Rashid.
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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