The French were 16-8 ahead at the break, but Stuart Lancaster's side led 24-19 with four minutes remaining, before the hosts triumphed with one final push.
France, desperate to erase memories of last year's championship when they managed only one victory, had stunned England when winger Yoann Huget touched down for the first try after 30 seconds.
England coach Lancaster, who must now prepare his side for a trip to Murrayfield to face Scotland on Saturday, said: "At the start we put ourselves into a bit of a hole. I thought the boys showed incredible character, resilience and a lot of skill to put ourselves in a position to win it. But a bit of French flair at the end and they got the victory."
The kick-off was knocked on by Jack Nowell and Les Bleus spun the ball right where Jules Plisson chipped ahead. His kick bounced away from Mike Brown and into the hands of Huget who raced over. Owen Farrell and Jean-Marc Doussain swapped penalties as the opening quarter continued to be played at full throttle. Disaster struck England for a second time in the 17th minute when flanker Tom Wood had the ball stripped from him in the middle of the pitch in an act that was the prelude to France's second try.
There seemed to be little on, but a weak tackle by substitute Alex Goode on Huget allowed the wing to offload to Brice Dulin. The full-back chipped ahead and once again the bounce deceived England, enabling Huget to score.
Wood was at the centre of a large scuffle as France took exception to a tackle that was marginally late, but no meaningful punches were thrown and the Northampton blindside escaped punishment. A collapsed scrum enabled Doussain to add another penalty before Goode danced free of Dulin only to knock on.
Finally, five minutes from the interval, England's attack clicked and a quick break from scrum-half Danny Care and good work from Billy Vunipola set up Brown for his first international score.
The visitors made a barn-storming start to the second half and Farrell landed a penalty before his side took the lead with Luther Burrell's 48th-minute try and the Saracens fly-half's conversion.
Vunipola blasted a hole through the midfield and offloaded out of the tackle to the debutant centre who had picked a superb line that led him straight over the whitewash.
A try-saving tackle by Billy Twelvetrees on Wesley Fofana kept France at bay, while at the other end a drop goal from Care extended England's lead to five points. Substitute Machenaud landed three points and Goode replied in the same vein. The game became a cagey exchange of kicks, each side seemingly waiting for the other to commit an error, but it was not a mistake that enabled France to break from their half and retake the lead.
Dimitri Szarzewski showed great awareness to delay his pass to Fickou and the wing stepped inside England's cover and sprinted in under the posts, before Maxime Machenaud's successful conversion left the stunned visitors holding their heads in despair.
France coach Philippe Saint-Andre said: "We were the first and the last to shoot, but we suffered physically in the second half, we missed a lot of tackles.
"There's a lot of work still to do, but what we'll have to remember is the victory, the smiles on the players' face after the game. It's a great start to the season."
France: Dulin, Huget, Bastareaud (Fickou 74), Fofana, Medard, Plisson, Doussain (Machenaud 57), Domingo (Forestier 48), Kayser (Szarzewski 43), Mas (Slimani 48), Flanquart (Maestri 43), Pape, Nyanga, Le Roux (Burban for Le Roux 41), Picamoles (Chouly 65).
England: Brown, Nowell (Barritt 65), Burrell, Twelvetrees, May, Farrell, Care (Dickson 61), Marler (M Vunipola 51), Hartley (T Youngs 58), Cole, Launchbury, Lawes (Attwood 67), Wood, Robshaw, B Vunipola (Morgan 65).
Referee: N Owens (Wales). Att: 80,000