Riding on Bell's half-century, England first posted a challenging 269 for six, despite some good bowling by the Australian bowlers, after electing to bat.
The hosts then returned to tighenten the noose around Australia's neck with a very disciplined bowling display and eventually restricted the visitors for 221-9 in their 50 overs at Edgbaston here.
Seamer James Anderson became the highest wicket-taker in one-day cricket for England during his three-wicket haul as he along with Tim Bresnan (2/45) and Stuart Broad (1/35) put Australia under trememdous pressure.
For Australia, skipper George Bailey was the top-scorer with a 69-ball 55, while James Faulkner scored an unbeaten 54 but they never got the support from the other end as Australia kept losing wickets at regular intervals.
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Earlier, Bell's partnership of 111 with Jonathan Trott, who made 43, was the backbone of the English innings as Bopara (46 no, 3x4, 1x6) struck a few meaty blows in the end.
During his 115-ball innings, Bell hit seven boundaries and seldom did he hit any rash stroke.
For Australia, seamers Clint Mckay (2/38) and James Faulkner (2/48) were the most successful bowlers.