However, captain Eoin Morgan put the onus of their defeat on their poor fielding.
Tournament favourites England, whose rise to the top of the one-day worldwide rankings since Bangladesh knocked them out of the 2015 World Cup has been based on aggressive batting, lived up to his words.
England, touted as favourites for the ongoing World Cup, suffered a 14-run loss to Pakistan at Trent Bridge on Monday.
Shakib too eventually departed for 121, unable to play a yorker from Ben Stokes that went on to hit the stumps and signaled the end of Bangladesh's resistance.
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Archer, whose 10 wicketless overs against Pakistan cost 79 runs, was far more threatening on Saturday as he reached speeds of 95 miles (153 kilometres) per hour.
The only downside for England was a hip injury to Jos Buttler, which will be assessed on Sunday.
Bangladesh shocked the cricketing world with a win against South Africa in their first match, but lost their second to New Zealand.
Pace bowling all-rounder Stokes followed up to finish with figures of 3-23.
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Then, Bangladesh pacer Mashrafe Mortaza broke the partnership in the 19th over as Mehidy Hasan caught Bairstow at cover-point. Watching the ball dribble for four because of a misfield, Roy - jogging leisurely between the wickets - accidentally ran straight into unsuspecting West Indian umpire Joel Wilson.
"There was a bit of change in the game plan". While their bowlers went for plenty conceding 348 runs, the England batsmen also failed to come out with an attacking performance which has been associated with them over the last two years.
A shaken Wilson took a while to get to his feet, as England's players looked on from the balcony of the team's dressing room with laughter and then in apparent concern.
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Bangladesh had won the toss and chose to field.