England cruised home with 29 balls to spare in the first match of the series at Trent Bridge, comfortably overhauling Bangladesh's 250 for nine.
Bell, recalled for the series as a replacement for the injured Kevin Pietersen [ Images ], made the most of his opportunity with a confident knock.
Roqibul Hassan top-scored with 76 and Junaid Siddique hit 51 for Bangladesh, whose total was their second-highest against England in one-dayers.
Roqibul was making his first start since returning to the side three months after he retired from all cricket in March, then changed his mind a week later, only to be banned by the Bangladesh Cricket Board until June 30 for upsetting the team's plan.
James Anderson [ Images ] was plundered for 74 runs from his 10 overs although the pacemen did claim three wickets.