Adding the ODI captaincy to his Test duties following Aaron Finch's 50-over retirement last month, Cummins made an impressive start on Thursday as Australia went 1-0 up in the three-match series.
After winning the toss and electing to field first, Australia kept England to 9-287 despite a majestic century from Dawid Malan.
England's total always appeared below-par as Australian pair David Warner and Travis Head (69) came out blazing with a 147-run opening stand.
After being left out for all but one game of the recent Twenty20 World Cup campaign, Steve Smith made a welcome return to form with an unbeaten 80 to allow Australia to cruise home with 19 balls to spare.
Smith's innings came at better than a run-a-ball as the star batter built on the good foundations set by Warner and Head.
Emerging allrounder Cameron Green pressed his case for a regular ODI spot, finishing unbeaten with Smith on 20.
Warner appeared set to score his first international century since January 2020 but was caught in the deep for 86.
Earlier, Malan responded to missing England's T20 World Cup triumph by smashing a career-best 134.
The 35-year-old's second ODI century helped England to a respectable total after the visitors found themselves in early trouble at 4-66.
England opted not to risk Malan for Sunday's T20 World Cup final against Pakistan at the MCG as the left-hander recovered from a groin injury.
But the No.3 played a lone hand for the tourists against an Australian attack determined to bounce back from a disappointing early World Cup exit.
Malan offered a chance when he was on 33, smashing a cut shot, but a diving Ashton Agar put down what would have been a classic catch.
Having reacted quickly to run out Liam Dawson on 11, the Australian allrounder had another highlight late in the innings, stopping what looked a certain six when he dived over the boundary and flung the ball back into play.
Malan was out two balls later when he skied a shot off Adam Zampa (3-55) and was caught at long-on by Agar.
Cummins (3-62) removed opener Phil Salt and No.4 James Vince in quick succession to have England reeling, while Mitchell Starc bowled Jason Roy with a terrific in-swinger.
But Malan was able to drum up enough support to rebuild England's innings.
This three-game ODI series started just four days after England's World Cup win against Pakistan.
England captain Jos Buttler was one of only three players to back up from that decider on Thursday, along with Salt and quick Chris Jordan.
The star of England's second T20 World Cup triumph, Ben Stokes, will not be playing in this series after his shock call in July to retire from ODIs.
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