After being dropped by Gary Ballance when he was on just eight, Finch continued to blast 121 runs from 128 balls as Australia won with 4.2 overs to spare.
Finch rode his luck early in his innings but quickly seized control, racing to 50 from just 47 balls and outpacing opening partner David Warner.
Finch and Warner put on 163 for the opening wicket but not without some controversy.
Warner was given a reprieve by the television umpire when he was on 22 despite the Australian opener accepting that a low catch claimed by wicketkeeper Jos Buttler had carried.
Warner, who had walked to the boundary, was recalled to the middle and went on to make 65.
He was eventually dismissed holing out to Ed Stokes on the boundary off part-time spinner Joe Root, and when Chris Jordan bowled Shane Watson for a duck two runs later England had a slim hope of pegging Australia back.
But Finch and captain Michael Clarke (43) quickly took the game away from England with a 72-run partnership, which ended when Finch sliced a Stokes delivery to Ballance at third man.
Earlier, a patient 79 from Ballance and a whirlwind 50 from Eoin Morgan helped England to post what had looked like a competitive total.
Ballance, who came to the crease when England were struggling at 22-2, held the innings together before accelerating towards the end when quick runs were needed.
He shared in an 83-run partnership with Morgan, who blasted his 20th ODI half-century off just 47 balls with five fours and a towering six.
England had earlier won the toss, with skipper Alastair Cook having no hesitation in batting first on a flat MCG wicket.
But Cook’s miserable tour of Australia continued when he fell for four, edging Clint McKay to wicketkeeper Brad Haddin.
McKay struck again to remove the equally out of form Root, who was trapped in front leg before for three with the score on 22.
Ian Bell (41) tried to launch a rescue mission, hitting two fours and a six off Glenn Maxwell, but with the score on 62-2 he went for another heave over midwicket and was clean bowled by left-arm spinner Xavier Doherty.
That wicket brought Morgan to the crease and the Irishman played superbly, showing a confidence missing from the English Test players over the Ashes series.
Some big hitting late in the innings from Buttler (34) and Tim Bresnan (16) allowed England to reach a defendable total, but it was not enough against the rampant Australian batting line-up.