Roy struck England's highest one-day score to guide his side to victory.
Roy blasted 16 fours and five sixes in a fabulous 151-ball innings, registering England's highest individual score in ODIs and surpassing the previous record knock at the MCG, Mark Waugh's 173 in 2001.
Fast bowler Andrew Tye debuts for Australia and captain Steve Smith, fresh from a remarkable Ashes series with the bat, said he wanted to see an improvement from his team in one-day cricket.
England skipper Eoin Morgan won the toss and elected to bowl first in Melbourne.
He was out soon after, mis-hitting a ball from Moeen Ali which was caught by Jonny Bairstow at wide 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.
Finch and Marsh, who made 50, fell in quick succession and it was left to Marcus Stoinis and Tim Paine to provide Australia's late assault and give them a defendable total.
Roy teed off early on against Mitchell Starc and Pat Cummins, showing a willingness to "chance his arm" in the words of the Australia captain Steven Smith that epitomises the daring of England's limited overs set-up since they started from scratch in the aftermath of a dire 2015 World Cup campaign.More news: Nintendo Switch Getting Remake of Classic DS RPG
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Mark Wood and spinners Moeen Ali and Adil Rashid had impressed with the ball earlier in the day, the former operating at a brisk pace on his return to global action.
The home side set England a target of 305 in the first of five One Day Internationals, a modest target made to seem all the more average by Roy's marvellous century.
Roy and Joe Root (91 not out) added 221 runs for the third-wicket as England romped to victory with seven balls to spare.
Roy sailed past his hundred during the incredible stand, while Root brought up his fifty.
England captain Eoin Morgan said the return of Roy, who missed selection in the Champions Trophy past year, and Root's shift down to No. 4 had really lengthened the batting lineup.
"It just looks like the way that they play is everyone just goes really hard and Joe Root is sort of the rock in the middle", Smith said. "They got off to an absolute flyer", Smith said.
Roy survived being given out and several close calls before eventually being caught for 180 to help guide his side to a five-wicket success to kick-off the five-match series in winning fashion. "It sums up the character of the bloke, coming back in".