While Wales' hopes of a successful title defence are over following defeats against Ireland and England, France remain unbeaten in the tournament.
France needed little more than two minutes to open their account as full-back Melvyn Jaminet slotted a penalty from 40 metres.
But Les Bleus were immediately guilty of obstruction under the high ball for Wales captain Dan Biggar to level matters at 3-3.
A scrappy start was ignited by France punishing a sloppy Wales kick out of defence, and the ball was recycled through several pairs of hands for Jaminet to give Anthony Jelonch a try-scoring pass.
Jaminet converted for a 10-3 lead and Wales suffered another blow as scrum-half Tomos Williams left the field with a head injury.
Wales responded to that double setback by sending Alex Cuthbert clear down the right flank.
The winger kicked on and excellent follow-up work from Taulupe Faletau won a penalty for Biggar to reduce the deficit to four points.
Wales had reversed France's early momentum but several promising opportunities were wasted as English referee Matt Carley punished their indiscipline.
But Biggar set up good field position and the fly-half dispatched his third penalty to leave France leading 10-9 at the interval.
Wales continued to fall foul of referee Carley after the break and Jaminet extended France's lead to 13-9 with a straightforward kick.
Wales failed to take a gilt-edged chance when Faletau gathered Biggar's clever kick near the touchline.
Faletau off loaded to Davies in the same movement, but the centre could not make the pass stick with the try-line beckoning and Dupont was in the right place to clear.
With the game opening up and still only one score between the sides, Wales sent on British and Irish Lions speedster Louis Rees-Zammit for the final 15 minutes.
France had to withstand some Welsh pressure in the final 10 minutes but held on for a hard-fought 13-9 victory.
Wales captain Dan Biggar felt France were fortunate to win as the reigning Six Nations champions lost for the third time in four games this season.
Biggar said: "It's very disappointing. I thought we were the better team for large periods of the game and one or two big moments is what Test matches hinge on.
"We didn't quite nail those moments and we paid the price.
"I'm so proud of the way the lads stuck to it against probably the form team in world rugby at the minute. That's a bit more like us in the terms of attitude, we didn't show that in the first half against England and Ireland, but ultimately we didn't come on the right side of it."