But Finch reached his ninth ODI century - his fourth against England - and Marsh (50) posted his 10th half century in a partnership which first restored the balance, then gave Australia the upper hand.
Jason Roy struck England's highest ever one-day global score as the tourists ended their long wait for a win in Australia with victory in the first ODI in Melbourne.
Root was unbeaten on 91 at the end as England reached 308-5 in the 49th over, having joined Roy in the sixth over when England was 60-2.
The successful chase was also a record for a one-day worldwide at the MCG, eclipsing Australia's 297 for four against England in 2011, and vindicated captain Eoin Morgan's decision to bowl first.
Roy scored 180 runs - the highest ever ODI score by an English batsman - as his side chased down Australia's target of 305.
When he reached 75 he surpassed 3,000 ODI runs and he went on to a century from 112 balls, raising the milestone another six off Rashid, the third of his innings.
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With England cruising to a comfortable win, Australia finally got the breakthrough they so desperately needed as Starc removed Roy for a career-best 180, which came off 151 balls and included 16 boundaries and five sixes.
Marsh was bowled by Rashid as he stepped away to leg and missed a sliding delivery that hit leg stump.
Pat Cummins and Mitchell Starc picked up two wickets each.
England suffered a few late hiccups with the quick dismissals of captain Eoin Morgan (1) and Jos Buttler (4).
"We knew the boys were going to be a bit down and it was our task to come in and light up the mood, bring a bit more positivity and get the boys going really well", Roy said.
"We've been losing too many wickets in those middle overs and not getting the partnerships together, and it's something we need to work on". It shows the character of the bloke.
"We've got to sort out our one-day cricket, it hasn't been as good as we would have liked over the past 12-18 months". I thought it was incredible from Jason - when he scores runs he's quite imposing at the crease.