The hosts' Nicolas Mahut and Julien Benneteau defeated Jamie Murray and Dominic Inglot 7-6 5-7 7-5 7-5 in Rouen, eliminating Great Britain without the need for two scheduled singles rubbers on Sunday.
Dan Evans and Kyle Edmund had previously lost singles matches on Friday, meaning Saturday's doubles was a must-win for Great Britain but they were ultimately unable to keep the quarter-final alive into the final day.
France, who haven't won the Davis Cup in 16 years, asserted their dominance immediately on Saturday when Mahut and Benneteau snatched the opening set on a tie-break. Across the entire tie, it was France's seventh consecutive set without Britain winning any.
The rot was stopped, albeit briefly, by Britain's doubles team when Inglot struck a majestic lob to gain control of the second set allowing Murray to serve his side onto the score-board. It would prove to be Britain's finest moment of the quarter-final.
Mahut, a former world No 1 at doubles, showed his class in the third set moments after wasting a set point by forcing another advantage for France, which they took to go 2-1 ahead.
Mahut played a shot from the crowd's side of the advertising hoardings in the stand-out moment of the final set, which France deservedly took to confirm a semi-final.