England, the 2003 champions, had knocked the French out in the semi-finals in the last two tournaments but had no answer to Les Bleus in the first-half as Vincent Clerc and Maxime Medard scored first-half tries.
France, who had so much to prove after losing to Tonga in their final pool game, led 16-0 at the break but Ben Foden gave Martin Johnson's side a lifeline with a 54th-minute try and Mark Cueto's score four minutes from time set up a grandstand finish.
But they were unable to get the converted try required to force extra-time and it is France who are through to a semi-final against Wales next Saturday.
England had won four of their last five matches against France but trailed after 11 minutes as Dimitri Yachvili nailed his first penalty after Toby Flood was called for not releasing.
Matt Stevens was then penalised for collapsing the scrum and Yachvili extended the advantage before Clerc scored his 11th World Cup try on 21 minutes after some woeful English defence.
Julien Bonnaire stole the lineout and winger Clerc burst through a gap between two defenders before spinning out of a weak Foden tackle to make it 11-0.
Few were expecting much flowing rugby before the game but, eight minutes later, the French had scored another try through their backs.
Thierry Dusautoir found a massive gap in the English defence and, although he was held up just short, Marc Lievremont's side worked the ball out to the left touchline and Alexis Palisson dumped off inside to Medard to score France’s second try.
England had their best moments of the opening period in the closing seconds but handling errors cost them dearly with Cueto knocking on just short of the line and Chris Ashton unable to take a high pass with the try line beckoning.
England finally came alive in the second-half and a quick tap penalty by Ben Youngs was the catalyst for an attack that concluded with Foden cutting back inside to nip through the defence and score.
A drop goal from replacement Francois Trinh-Duc took France's lead to 12 points with eight minutes remaining but Cueto gave England hope as he touched down, under pressure from Clerc, with the try given after a lengthy TV referral.
Flood sliced the conversion to leave England needing a converted try in the last three minutes to force extra-time.
But Manu Tuilagi was dragged into touch, France took possession from the lineout, gained ground with the maul and England were penalised allowing the clock to run out as Morgan Parra prepared for his penalty attempt.
His effort hit the upright but it was incidental as France proved that the cliche about wounded animals being most dangerous could have been written about this side.
10' Yachvili penalty - France 3-0
15' Yachvili penalty - France 6-0
21' Clerc try - France 11-0
30' Medard try - France 16-0
54' Foden try - France 16-5
55' Wilkinson conversion - France 16-7
72' Trinh-Duc drop goal - France 19-7
76' Cueto try - France 19-12
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