Wales were down by 17-8 at half time but had been kept within reach of an older, and much more physical French side with a well constructed try for right wing Ryan Evans and a Sam Davies penalty.
Davies, who had kicked the winning drop goal against England last Saturday, had an uncharacteristic off day with the boot, missing 11 point in total, including a last minute penalty attempt.
Despite that, Wales fought their way back into the game, winning the second half 10-3 with a second Ryan Evans try and a conversion and penalty to Davies.
Team manager Roy James said, "We are extremely proud of the boys. They tested the French, which is something that hadn't been done by either of the other sides in the tournament. They really did us proud today and throughout the tournament."
Welsh openside Daniel Thomas was voted player of the tournament by all head coaches.
James added, "Daniel was outstanding in all three of our games and I'm sure this tournament will stand him in good stead for his rugby future along with many of the other players."
Wales scorers: Tries: Ryan Evans (2); Con: Sam Davies; Pens: Sam Davies (2)