Australia were 2-121 at that stage after Aaron Finch fell in the seventh over to Chris Jordan for 22 off 20 balls.
But Warner's dismissal didn't slow Australia's momentum with Shaun Marsh picking up where the swashbuckling opener left off.
Marsh finished with 71 off 89 balls. Brad Haddin was the other not out batsman on 37 off 33 balls.
Earlier Eoin Morgan topscored with 54 off 58 balls in an England innings that never really got going.
Morgan fell in controversial circumstances when Dan Christian took a great catch inches off the turf off his own bowling in the 42nd over.
The Irishman stood his ground, sparking a war of words between Morgan and non striker Jos Buttler and Australian captain Michael Clarke in scenes reminiscent of Clarke's stoush with James Anderson during the Test series.
Eventually the third umpire ruled Christian's catch legitimate and Morgan was on his way.
Christian's catch was emblematic of a fantastic fielding effort from Australia.
Morgan's dismissal left England on 6-186 before some impressive late hitting from Tim Bresnan (41 off 29) pushed the visitors beyond 240.
Nathan Coulter-Nile was the best of the Australian bowlers with 3-47 off his 10 overs.
James Faulker and Christian both claimed two wickets a-piece.
In his first international game since the Ashes in England in July last year, and his first ODI since September 2012, James Pattinson managed 0-41 off his six overs.
Australia's win continues their dominance of England this summer after their 5-0 Ashes victory.
Clarke said a great effort in the field paved the way for the win.
"We did fantastically well in the field," Clarke.
"The Australian way is to lead the way in the field. When we are fielding well we are playing well.
"But our attitude hasn't changed, we still want to win the series five-nil."
England were 0-50 after 50 balls before Cook fell and the England captain said his side again failed to take their chances.
"The first 10 overs we played alright but after that we kept losing wickets and lost momentum," he said.
"Every guy got 20 or 30 but you have to go on with it to win the game.
"Obviously Australia has won a lot of cricket against us in the last few months but we have to try and win a game."
The fourth game in the series is in Perth on Friday.
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- AAP