The Welsh prevailed thanks to a try by Liam Williams and three penalties by Leigh Halfpenny to end the championship behind Ireland who sealed the grand slam by beating England 24-15 at Twickenham.
France, who have been showing signs of improvement since they beat England last Saturday, scored a try through Gael Fickou with Francois Trinh-Duc kicking a drop goal and Maxime Machenaud slotting home a conversion and a penalty.
Les Bleus had the upper hand, notably in the second half, but finished the tournament in fourth place in new coach Jacques Brunel's first campaign in charge.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"It was definitely an ugly win. Sometimes you will take an ugly victory over the performance. We turned up today and it was all about winning," Wales coach Warren Gatland said.
"The performance was not the most important thing. We started off well but there were too many turnovers and mistakes," he added.
"That is not a bad French team and they are going to get better. It is a tough competition to win or finish second and we are pleased with where we are at the moment."
The visitors started well and Trinh-Duc put them ahead with a fine drop goal.
But they made a mistake on Wales's short kickoff, letting the ball bounce to allow the home side to grab possession.
Scott Williams kicked the ball through, Trinh-Duc could not control it and Liam Williams dived over to put his team in front.
Halfpenny missed the conversion but added three points with a penalty.
France were penalised again -- the third time in the opening 15 minutes -- and Halfpenny extended the lead to eight points.
The visitors hit back when Fickou, who started a nice collective move, ended it with a try after Machenaud, who converted, and Trinh-Duc played quickly to send him over.
Les Bleus, however, suffered a setback when flanker Yacouba Camara, one of their best players in the championship, was replaced by Mathieu Babillot after hurting his left knee.
Another Halfpenny penalty put Wales four points ahead as the home side benefited from France's lack of discipline.
Brunel's side dominated and were rewarded with a penalty converted by Machenaud early in the second half.
They enjoyed possession but Wales defended ferociously and Trinh-Duc, who took over the kicking duties after Machenaud was replaced, missed a routine penalty as the visitors failed to score when they had the chance.
France had one last penalty in the dying minutes but they lost the resulting lineout and had to settle for a frustrating defeat.
(Writing by Julien Pretot; Editing by ED Osmond)
