The first half was a fairly even affair - Yoram Moefana scored for France and Darcy Graham replied for Scotland. Three penalty kicks from Thomas Ramos and two from Finn Russell meant the hosts led 16-13 at the break.
The Scots could have gone into the break ahead but Tom Jordan's try was ruled out as Blair Kinghorn had been in touch earlier in the move.
Jamie Ritchie and Jean-Baptiste Gros were yellow-carded while Peato Mauvaka somehow escaped with a sin-binning after an assault on Ben White after play had been stopped.
Scotland started the second half brightly, but Louis Bielle-Biarrey was first to score as France took advantage of a loose pass.
Ramos scored France's third try in the 58th minute as they started to pull clear on the scoreboard. Moefana got his second try just after the hour mark securing a 4-try bonus-point for Les Bleus.
The hosts were now in control but determined defence from Scotland stopped France adding to their score as they secured a well-deserved seventh Six Nations title.