Australia, who conceded a world-record 481-6 to England in their heaviest-ever defeat at Trent Bridge two days ago, regathered to post the highest ODI total at the ground.
The tourists built steadily after captain Tim Paine chose to bat with Finch (100) and Travis Head (63) putting on 101 in 19.1 overs.
Head, whose nine boundaries included some glorious on-drives, couldn’t hide his disappointment after pulling Adil Rashid straight to David Willey in the deep.
Finch made no such mistake, though, sharing a second successive century stand with Marsh and reaching his 11th ODI ton and sixth against England off 105 balls with six fours and three sixes.
The opener had barely finished celebrating when he fell to his very next ball, trapped lbw by a Mark Wood delivery that skidded onto him to leave the tourists on 225-2.
Wood claimed a second victim four balls later as Marcus Stoinis went back to a full delivery and was bowled for just one.
Marsh, who reached his 50 with a six over long-on off Moeen Ali, clobbered Willey back over his head for six at the start of the 45th over to launch the charge and plundered 25 off one Rashid over, bringing up his fifth ODI century with a six over midwicket.
It took a supreme partnership catch to end his knock – Marsh launching Willey to deep midwicket where Roy took the ball before jettisoning it to Overton before he stepped over the rope.
That was the second of three wickets to fall to Willey in the 48th over – Carey picking out Overton at midwicket off a full toss before Neser (two) tickled a catch behind.
Willey added the wicket of Paine, trapped lbw, in the final over to finish with 4-43. Craig Overton, making his ODI debut in place of the rested Liam Plunkett, was the most expensive of the bowlers with 0-55 off seven overs.
Watch England’s chase live on Sky Sports Cricket.
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