England found the French too hot to handle in the first half and trailed 16-0 at the break after tries for Vincent Clerc and Maxime Medard.
The Red Rose, who included Dan Cole, Tom Croft, Louis Deacon, Toby Flood, Manu Tuilagi and Ben Youngs in their team, fought back in the second-half.
Tigers scrum-half Youngs tap-and-go provided the spark and Ben Foden added the finish to put England’s first points on the board after 55 minutes.
And when Mark Cueto latched onto Flood’s kick through for his fourth try of the tournament with three minutes remaining, England needed a converted try to take the game into extra-time.
However, France had the composure to close the game out and book a semi-final against Wales, who had earlier beaten Ireland 22-10.
Horacio Agulla starts on the wing for Argentina in their quarter-final with hosts New Zealand. His Tigers teammate Marcos Ayerza is on the bench for the game in Auckland (8.30am BST).
The other quarter-final sees Australia take on South Africa in Wellington (6am BST).
