Chasing 281, England were stunned by Mitchell Starc snaring Jason Roy and Dawid Malan in the opening over while they stumbled to 34-3 before Vince and Billings put on 122 to steady the ship.
But Vince's downfall for 60 off 72 balls opened the floodgates, with Billings departing for 71 off 80 deliveries soon after, as England were all out for 208 in 39.5 overs to lose by 72 runs.
A dead rubber awaits in Melbourne on Tuesday, an ill-fitting conclusion to what has been a memorable past few weeks Down Under for England, who just last Sunday were celebrating their T20 World Cup win.
Steve Smith's 94 off 114 balls underpinned Australia's 280-8, with Marnus Labuschagne amassing 58 off 55 deliveries and Mitchell Marsh 50 off 59, which proved more than enough on a wearing surface.
The timing and relevance of this series has been questioned and both sides were without their captains as Moeen Ali replaced Jos Buttler and Josh Hazlewood led Australia in the absence of Pat Cummins.
Joining Moeen, leading England for the first time in an ODI having skippered the T20 side on 11 occasions, were fellow World Cup winners Sam Curran, Chris Woakes and Adil Rashid.
Moeen's first major decision involved bringing himself on to bowl after Australia had raced to 33 for nought in five overs. The change worked as David Warner swept to Liam Dawson inside the ring.
Woakes tightened up his lines and varied his pace after starting loosely, but it was a quicker ball banged in that did for Travis Head, whose pull caught the splice and looped to short midwicket.
Smith bedded in well, content to take the singles on offer, while Labuschagne showed more intent, evidenced by whipping his second ball for his first ODI six after advancing down the track to Moeen.
He used his feet well to England's spinners but an attempted sweep caught the top edge and ballooned to mid-off, while Rashid had two in two when Alex Carey was stumped as Australia slipped to 144 for four.
Smith upped the ante after reaching a 70-ball half-century - his fourth successive fifty-plus score in this format - but holed out to Rashid, who finished with three for 57, while Woakes' knuckle ball castled Marcus Stoinis on a pitch that was slowing down and showing signs of reverse swing.
Marsh clubbed legside sixes off Woakes and the off-colour Curran, but he and Starc were caught on the boundary off successive David Willey deliveries.
But Australia finished well, with Curran, player of the tournament at the T20 World Cup, conceding 16 in his final over to finish with 7-0-55-0.
England made a false start at the outset of their reply as Roy gloved down the leg-side for a two-ball duck, his fourth in his last eight innings for club and country, while worse was to follow when Malan, a centurion in Adelaide, had his off-stump knocked back by a sensational Starc outswinger.
Phil Salt bristled with purpose but exposed his stumps to Hazlewood and paid the price after missing a waft to depart for 23, yet Australia removing Starc after three overs let Vince and Billings settle.
The pair are on the fringes of this white-ball set-up but both have extensive experience at the Big Bash League, with Vince particularly fluent through his favourite cover region, an area he cleared with a fantastic drive off slow left-armer Ashton Agar, while Stoinis was also deposited into the stands.
Billings settled by rotating the strike for his first 25 runs but reverse swept Adam Zampa for his first four then shuffled down the wicket to fellow spinner Agar and unleashed a couple of mighty blows for back-to-back leg-side sixes.
The pair brought up their half-centuries and took England past 150 before Hazlewood reintroduced himself and snared Vince, struck on the back leg and given lbw after missing a heave across the line.
Moeen seemed to have Zampa's number with swipes for four then six but a regulation leg-break beat his defences, sailing through the gate.
In Zampa's next over, Billings yorked himself and was bowled which ended England's hopes. Both Starc and Zampa finished with four wickets as Australia won comfortably.