Roy showed his intent from the first ball he faced and he got his rewards by surpassing Alex Hales, who held the record for the previous highest score by an England batsmen with 171.
Roy was supported well by Joe Root, who finished 91 not out as he tourists chased down 305 to win the first match of the series.
Morgan unsuprisingly decided to bowl first in Melbourne
On a beautiful sunny day in Melbourne, England captain Eoin Morgan won the toss and chose to bowl first giving the fact his team like to chase.
Three early wickets left the Aussies on the back foot
Morgan's choice was vindicated early on as Mark Wood, returning to the side after injury, removed the dangerous David Warner for two early on.
Steve Smith joined Aaron Finch at the crease at the fall of the Warner wicket and just as the pair started to build a solid partnership, Adil Rashid struck to remove Smith for 23.
Not long after that, Liam Plunkett then got into the action as he removed Travis Head for five to reduce the hosts to 78-3.
Finch brought up his ninth ODI ton for the hosts in a brilliant partnership with Marsh
Finch though was still at the crease and alongside Mitchell Marsh, the pair put on a brilliant century partnership to give the Aussies a decent platform.
During the partnership, Finch also brought up his ninth ODI hundred for Australia with a big six which was his third hundred against England in a row at his home ground.
England needed some inspiration to break the partnership and it finally came when Moeen Ali removed Finch for 107 to give the tourists a much-needed breakthrough.
Marsh though soon after that wicket brought up his own half-century before falling two balls later off the bowling off Rashid for 50.
Late hitting from Stoinis and Paine get the Aussies up past the 300 mark
That meant that the Aussies needed Marcus Stoinis and Tim Paine to rebuild the innings towards the end and the pair did just that as they put on 80 for the sixth wicket.
Stoinis smashed a 37-ball half century during the partnership and just as he was looking to keep going, he fell off the bowling of Chris Woakes for 60 to give the tourists a crucial breakthrough.
Plunkett then removed Paine for 27 and Pat Cummins for 12 in the final over the innings as the Aussies made 304-8 from their 50 overs which meant that the tourists needed to produce a record chase in Melbourne to win the opening game of the series.
Roy and Bairstow got the tourists reply off to a rapid start before losing two quick wickets
In reply, the tourists got off to a very fast start as Roy and Jonny Bairstow put on a half-century stand for the first wicket in double quick time.
Just as the pair tried to keep going though, Mitchell Starc got the breakthrough when he removed Bairstow for 14.
Not long after getting the first wicket, the Aussies got a second when Cummins removed Hales for four to reduce the tourists to 60-2.
Roy brought up his fourth ODI century by smashing the ball to all parts of the MCG
Roy though continued on his merry way despite the two quick wickets by playing loads of shots and he brought up his half century from 32 balls.
Alongside Roy, Root supported the opener well as the pair took their partnership passed the century mark.
During the partnership, Roy also brought up his fourth ODI century for England after receiving a reprieve on 91 when he a reviewed an LBW decision which saw the umpire give him out.
Root hit yet another half-century in a double century partnership with Roy
Root also looked good at the crease as he cruised to another half-century as the pair put on a double century partnership for the third wicket. Roy also passed the 150 mark as the Aussies bowlers just couldn't find a way to break the fantastic partnership.
Roy broke the England batting record in ODIs as the tourists claimed a five-wicket win
Then the big moment came for Roy soon after when he went past the highest individual score by any England batsmen in ODIs.
Roy though did finally fall before the end when Starc removed him for 180 to end a brilliant partnership.
The Aussies then removed Morgan for one and Jos Buttler for four prior to England reaching the target, before Root, who finished the game 91 not out and Ali saw the tourists to victory with seven balls to spare to win by five wickets.