England skipper Eoin Morgan and Ben Stokes seemed amused by the incident on the field.
Morgan made 35 before being caught and Plunkett and Woakes then blasted 45 off the last 17 balls to help England post the seventh highest total in a World Cup.
The Surrey batsman faced 121 balls, hitting 14 fours and five sixes as he fell just short of Andrew Strauss's England World Cup record of 158 against India in Bangalore in 2011.
But Shakib anchored the innings, flicking Archer for six on his way to a 95-ball century that also included nine fours.
"Probably the slow nature of the pitch made it more hard to play", said Morgan of the two fast bowlers.
The victory will help to exorcise the ghost of 2015 World Cup when England were knocked out by Bangladesh after beating them by 15 runs. A doctor came onto the field to carry out a concussion test on Wilson, who also went off the field during a drinks break for further checks.
Roy brought up his 150 with those three successive sixes but lost his wicket as he went for a fourth, mis-hitting Mehedi Hasan to cover. By the time he holed out to Soumya Sarkar on the boundary rope, Buttler was moving gingerly after hurting his right hip while sprinting a single.
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From there on, he appeared to be hindered while running between the wickets.
The English opener was looking at the ball and had no clue about the umpire's position.
Farbrace, speaking to BBC's Test Match Special, said: 'Roy could easily open for England in the Ashes, he has shown his temperament has changed.
Shakib took Bangladesh across the 200-mark in 37th over but the Tigers never really looked in the chase and once he departed in the 40th over, it was a mere formality thereafter for the hosts.
"You can think of chasing something like 320, 330, if you keep wickets in hand", Shakib said.
After opening with a win over South Africa, Bangladesh has lost to New Zealand and now England but gets a slight respite from its tough-looking early schedule when it plays Sri Lanka in Bristol on Wednesday. "So we'll use every day as much as we can to try and get him back on the park, and if that's not the case and he is a risk going into the next game, it might not be worth it". England's next game is against West Indies on Friday in Southampton.