Morgan, England’s new one-day captain following the jettisoning of Alastair Cook, cracked an excellent 121 to help his side bounce back from a horror start and post a total of 234 all out.
But Warner (127) bashed the Baggy Greens to a bonus-point victory, with the opener’s 18 fours ensuring the hosts chased down 235 within the required 40 overs and one ball.
Morgan elected to bat after winning the toss in Sydney, though he would have looked on in horror when both Ian Bell and James Taylor were back in the hutch just three balls into the match.
Bell (0), preferred to Alex Hales having scored 187 in England’s warm-up game against the Prime Minister’s XI, departed for a golden duck, trapped lbw by Mitchell Starc’s in-swinging first delivery.
Taylor (0) lasted just two balls before he also had his pad clattered by Starc and worse was to come for the tourists when they sunk to 12-3 after Joe Root (5) nicked off to Shane Watson in the slips.
Moeen Ali (22), who had smoked Starc for a 75-metre six, and Ravi Bopara (13) were then caught out by Glenn Maxwell, meaning England were 69-5 midway through the 16th over.
However, Morgan built partnerships of 67, 34 and 56 with Jos Buttler (28), Chris Woakes (8) and Chris Jordan (17) respectively, bringing up his seventh ODI ton with a sumptuous four.
The 28-year-old’s 136-ball knock, which included 11 fours and three sixes, came to an end when he was pouched by Maxwell in the deep off Starc.
Starc then delivered his second golden duck of the day, cleaning up Steve Finn, to polish off England’s innings with the penultimate ball of the 48th over and record figures of 4-42.
England, however, were shorn of their premier paceman, with James Anderson not yet deemed fit enough to return to the side, though Stuart Broad did play after also missing the pre-Christmas series in Sri Lanka.
Anderson’s absence was keenly felt as Australia raced to 71-2 in the 13th over, with Aaron Finch (15 off 18 balls) and Watson (16 off 20) supporting Warner with brisk cameos.
Warner was not hitting cleanly at the start of his innings but smacked himself into rhythm during an 87-run stand with Steven Smith (37) for the fourth wicket that ended when the latter was castled by Moeen.
Moeen served as England’s frontline spinner with the visitors opting to leave out James Tredwell, a move that seemed even odder after Australia tweaker Xavier Doherty took 1-38 from his 10-over spell.
Woakes was the pick of England’s bowlers, registering figures of 4-40 and even plucking the wicket of Warner when the Australia match-winner holed out to Bell at cover point with eight runs still required, but George Bailey’s squad eased home in 39.5 overs.
England and Australia are joined in the Tri-Series by India, with the sides playing each other twice in a league format and then the top two squaring off in the final on February 1.
Australia are next in action against India in Melbourne on Sunday, with England tackling MS Dhoni’s men on Tuesday in Brisbane.