Australia scored 251 for seven with opener Shane Watson [ Images ] scoring a muscular 66 and George Bailey reaching 65.
Their total looked at least 30 short of a defendable one and Bell made the most of his opportunity at the top of the order since replacing the retired Kevin Pietersen [ Images ].
He was dropped on 38 but made no further mistakes before he was bowled missing an attempted cut off Australia captain Michael Clarke's [ Images ] first delivery.
Bopara, seeking to impress before the forthcoming test series against South Africa [ Images ], was untroubled before he ran himself out with only two runs required.
Eoin Morgan [ Images ] was there at the finish unbeaten on 43 after Clarke was awarded an lbw decision with his second ball to the England left-hander who had yet to score. Morgan immediately appealed and television replays showed he had got a faint inside edge.
England, the world's top-ranked test side, can overhaul Australia at the top of the one-day rankings if they complete a series whitewash.