Jason Roy's 153 took the hosts to an imposing 386 for six, leaving Bangladesh needing to surpass Ireland's total of 329-7 against England in 2011 -- the highest score by a team batting second to win a World Cup match.
But their chase got off to a bad start when fast bowler Jofra Archer dismissed Soumya Sarkar for two with a ball that clipped the top of the left-hander's off stump.
Archer reached speeds of 95 miles (153 kilometres) per hour with the wind behind him, roughing up Tamim Iqbal, who carelessly spooned a catch to cover off Mark Wood for 19, leaving Bangladesh struggling on 63-2 in the 12th over.
But Shakib anchored the innings, flicking Archer for six on his way to a 95-ball century that also included nine fours as he became only the second Bangladeshi, after Mahmudullah, to make a World Cup hundred.
Bangladesh were 179-4 in the 33rd over but still required a run-rate of more than 12 an over.
Shakib put on 106 with Mushifqur Rahim (44), a partnership that ended when Roy, who dropped a straightforward chance against Pakistan, held a good catch at point.
And 169-3 soon became 170-4 when Mohammad Mithun fell for a duck after edging an intended drive off leg-spinner Adil Rashid to wicket-keeper Jonny Bairstow.
The Test gloveman was behind the stumps after Jos Buttler suffered a hip injury during a typically dashing 64.
Shakib was eventually out for 121, bowled by Stokes, to leave Bangladesh facing near-certain defeat.
England were sent in to bat by Tigers captain Mashrafe Mortaza on a green-tinged pitch, but Roy and Bairstow (51) seized the initiative during a rapid first-wicket stand of 128.
All of Bangladesh's bowlers suffered as England repeatedly took advantage of two short, straight boundaries and some poor ground-fielding.
The in-form Roy completed a 92-ball century in bizarre fashion when, not looking, he collided with umpire Joel Wilson as he completed what should have been a single to take him to 97, only for Mithun at deep square leg to misfield a ball that went for four.
Both Roy and Wilson were unhurt.
Roy was finally caught by Mashrafe off Mehidy Hasan Miraz, holing out after hitting three towering sixes in a row off the spinner.
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 20ll.