It was a fiercely professional performance from England, who bided their time to counter fleeting spells of inspiration from the hosts and clinched the win with four overs to spare as Jamie Siddons's side were unable to sustain their challenge.
Alastair Cook won the toss and put Bangladesh into bat on a slow wicket, and the hosts were bowled out for 228 after Tamim Iqbal compiled a majestic 125.
England got off to a solid start as Cook scored 64 to get his tour up and running, but the tourists slumped from 73 without loss to 96 for three as the Tigers roared back.
Vice-captain Paul Collingwood came to the crease and steadied his side with typical composure and his 75, in conjunction with Eoin Morgan (33) and Matt Prior (30), steered England to victory.
Iqbal's imperious century came off just 120 balls and earned the Bangladesh opener the man-of-the-match award as his side were bowled out inside 46 overs.
Tamim's knock included 13 fours and three sixes, two of them in a frenetic bout of boundary hitting in the first 10 overs but, having been put down by Eoin Morgan at cover on only 10, Tamim also impressed with his restraint in the middle overs.
Graeme Swann was the star with the ball for England, as he completed his 10 overs with figures of three for 32, while Stuart Broad pitched in with two wickets for the cost of 46 runs.
Craig Kieswetter (19) was beginning to settle at the crease after an uncharacteristically subdued start, but he was stumped for a rash charge down the wicket.
Kevin Pietersen felt aggrieved as he was adjudged to have got an inside edge with Junaid Siddique taking the catch off Bangladesh skipper Shakib Al Hasan to dismiss the number three for just one.
Cook and Collingwood looked ready to dig in until the former was seen off by a poor leg before verdict that gave Naeem Islam a second success and left the tourists on 96 for three.
But Morgan and Prior ably assisted the unflappable Collingwood, after Cook's breezy opening stint set England off on the right footing.
After a promising start, Bangladesh faded out of the match in disappointing fashion, as the tourists establish a 1-0 lead in the three-match series.