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Ian Bell top-scored with 91 on his Warwickshire home ground and shared a second-wicket stand of 111 with county colleague Jonathan Trott (43).
But with England looking at a score in excess of 300, Australia's seamers pegged them back until an unbroken stand of 56 between Ravi Bopara (46 not out) and Tim Bresnan (19 not out) boosted the total.
This was the first match in a Group A also featuring New Zealand and Sri Lanka, who meet in Cardiff on Sunday.
In Group B, India has already beaten South Africa and the West Indies have defeated Pakistan.
The top two teams from each group will qualify for the semi-finals, with the final at Edgbaston on June 23.