Scotland got off to a disastrous start, first conceding a try to Romain Ntamack and then seeing Grant Gilchrist red-carded for a shoulder to the head of Anthony Jelonch.
France extended their lead when Ethan Dumortier went over, but it was soon 14 aside as Mohamed Haouas was sent off for head-to-head contact with Ben White.
Thomas Ramos extended the French lead after intercepting a Finn Russell pass, but the Scots began to edge their way back into the game, and Huw Jones touched down before the break.
Ramos added a three-pointer before Jones crossed the whitewash for his second try to increase the nerves for the French.
And when Finn Russell scored with 13 minutes remaining, Scotland had all the momentum, but France controlled the final stages and sealed the match when Gael Fickou went over for a bonus-point try.