Australia was reeling at 221/9 in 36.3 overs despite a counter-punching half-century by skipper Mitchell Marsh in the middle.
But Carey led them out of trouble and to a decent total of 270 all out with a one-sided partnership with Josh Hazlewood.
Carey remained the top-scorer for Australia with a 67-ball 74, laced up with eight fours and three sixes. He was finally dismissed by Olly Stone in the fifth over.
Earlier, after being put into bat first, the visitors lost their in-form opener Travis Head (29) early with just 46 runs on the board in the ninth over, which paved the way for Marsh’s onslaught.
The right-handed batter smashed six fours and three sixes on his way to a 59-ball 60 before eventually falling victim to Jacob Bethell in the 26th over, reducing Australia to 151/5.
Australia kept on losing wickets and eventually slipped to just one wicket away from bundling out on a modest total before Alex Carey turned the tide in their favor.
Brydon Carse was the star with the ball for England, picking up three wickets. He was followed by Bethell, Adil Rashid, and Potts, who bagged two each.
The 271-run target turned out to be sufficient for new-look England, who had slipped to 56/5 inside 10 overs.
Wicketkeeper batter Jamie Smith offered some fightback with a cautious 49 until falling victim to Hazlewood in the 31st over, leaving England in tatters at 159/7.
England’s batting tail was toothless against a disciplined Australia bowling attack as their remaining three wickets only yielded 43 runs, resulting in the home side being eventually booked on 202 with around 10 overs to spare.
Mitchell Starc led Australia’s bowling charge with three wickets. He was backed by fellow pacer Hazlewood, Glenn Maxwell, and Aaron Hardie, who took two apiece.