Set 276 to win, England were bowled out for 260 despite Jos Buttler's 52-ball 65, while Rubel Hossain claimed 4-53.
Bangladesh had earlier posted 275-7 in Adelaide thanks to 103 from Mahmudullah and Mushfiqur Rahim's 89.
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England go out in the group stages for the third time in five World Cups, while Bangladesh advance past the first round for only the second occasion.
"I'm gutted at the moment," said England skipper Eoin Morgan. "There'll be an inquest over the next few weeks as to what happened and what went wrong. Then we'll go from there."
Coach Peter Moores said he felt "hollow inside" and "hugely disappointed" following England's demise.
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Bangladesh's victory also secured Sri Lanka's place in the last eight, alongside Pool A winners New Zealand and Australia.
England, meanwhile, face a dead rubber against Afghanistan in Sydney on Friday.
A fourth defeat in five games is the latest episode in 23 years of World Cup failure, since they reached the final in 1992.
In losing to every Test-playing side in Pool A - their only win has come against Scotland - England have put in their worst showing since a first-round exit on home soil in 1999. Even then, they managed to beat defending champions Sri Lanka.
Sri Lanka, along with Australia and New Zealand, are three sides to have dished out heavy defeats to England in this tournament, but all three are ahead of Morgan's team in the world rankings.
Bangladesh lie above only Zimbabwe, Ireland and Afghanistan, yet have now beaten England in three of their past four meetings.
One of those was with home advantage in the last World Cup, but this time England wilted when chasing a manageable target on a blameless surface.
They seemed in control when Ian Bell and the recalled Alex Hales were compiling a second-wicket stand of 54, only for the Bangladesh pacemen to return and drag England to a standstill.
Hales flat-footedly wafted Mashrafe Mortaza behind and Bell, who had looked fluent, was stifled before edging a Rubel lifter on 63.
In the same over, Morgan pulled to long leg for a fifth duck in 11 ODI innings and, when James Taylor flashed Taskin Ahmed to slip, England had lost three wickets for 11 runs.
Joe Root looked calm before edging Mortaza behind to leave Buttler and the tail requiring 113 from 14 overs.
The wicketkeeper's clean striking dragged England back in it, with 38 required from 28 balls.
However, Taskin returned to find another edge and, when Chris Jordan was run out from the next ball, the game looked gone.
Chris Woakes (42 not out) continued the chase in the company of Stuart Broad and, after Tamim Iqbal put down a simple chance to reprieve the Warwickshire man with only 18 required, England had another chance.
But in the next over Rubel bowled Broad and James Anderson in the space of three balls to start delirious celebrations of Bangladesh's most famous win.
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That they had pulled off such an upset was largely thanks to the brilliance of Mahmudullah, who made Bangladesh's first World Cup hundred, and the impetus of Mushfiqur.
Despite 12 of the 15 previous games held in Australia being won by the side batting first, England opted to field and the decision looked a good one when Bangladesh were reduced to 8-2 and 99-4.
But Mahmudullah, happy to throw his hands through the ball, and Mushfiqur, excellent square of the wicket, steadied then took advantage of England's occasional errors in length to build a stand of 141.
Fielding and death bowling improved on the rest of the tournament left England with a target that appeared within their grasp.
But the desperate attempt at a run chase was entirely in keeping with the rest of their World Cup campaign.
Listen to highlights from Test Match Special's and Radio 5 live's 2015 World Cup coverage.
England were knocked out of the World Cup in dismal fashion as Bangladesh claimed a stunning 15-run win.