[Scorecard: Second ODI]
With this win, England have now taken a 2-0 lead in the five-match ODI series.
Batting first Australia had made 251 for seven in 50 overs. However, this score was not enough to defend for this bowlers as the hosts reached the target in the 46th over.
England's chase was built around Bell and Bopara's magnificent innings. Opener Bell hit 75 runs while number four batsman Bopara had 82 runs to his name.
The finishing touches of the chase were given by left-hander Eoin Morgan. He remained unbeaten again with a crucial 43. He had scored 89 not out in the first match.
Morgan was given out LBW by on-field umpire Richard Illingworth but the left-hander had the decision overturned by TV umpire Aleem Dar. However, there was controversy over this dismissal as the replays suggested that Morgan had not hit the ball.
Bell hit six fours and a six in his 113-ball innings. Bopara needed 85 balls for his 82 with eight boundaries.
Australia's fast bowler Brett Lee went wicketless in his 10-over spell. He gave away 58 runs. Clint McKay, Shane Watson and Michael Clarke were the wicket-takers for the visitors while the other dismissal came through a run out. Bopara was the man who was run out.
Bell and Bopara had a game-changing 90-run partnership for the third wicket.
Earlier, Australia were helped by half centuries from Watson and George Bailey. Opener Watson scored 66 while Bailey supplied 65 runs to the team total.
Steven Finn, Bopara, Graeme Swann and Stuart Broad had one wicket apiece while Tim Bresnan had two dismissals in his bowling spell.
Captain Clarke said they were outplayed by England.
"I think the scoreboard at Lord's (on Friday) probably hid how far away the teams were. I think England outplayed us at Lord's, it only said 15 runs but I think they outplayed us at Lord's and they certainly outplayed us here," Clarke said.
"We have got areas not only with the bat we need to improve but with the ball as well," he added.
Bopara collected the Man-of-the-match award. Now the teams will move to Birmingham for the third ODI on July 4.