The French were 10-0 down after 26 minutes, but tries by centre Wesley Fofana and fullback Maxime Medard together with eight points from the boot of Morgan Parra helped inflict Scotland's third successive loss.
The World Cup finalists have four points in the standings, two behind unbeaten Wales but having played one game fewer. The pair meet on the final weekend of the northern hemisphere competition in what is shaping up to be a title decider.
To win a fifth Six Nations in nine years, France will have to do it the hard way after their home match against Ireland on February 11 was postponed because of a frozen pitch.
It left Les Bleus with a gruelling schedule of four games on successive weekends and they started rustily against the Scots, who have now lost five tests in a row for the first time since 2004.
"We put in another massive effort and to come away with nothing, everyone is pretty down," Scotland flanker Ross Rennie said.
Debutant fullback Stuart Hogg's eighth-minute try gave Andy Robinson's side the lead, with Greig Laidlaw converting and adding a penalty in the 26th.
France bounced back through Fofana, who burst through the Scotland backline on a great angle with a powerful run that took him through several tacklers and over the line.
Parra converted and knocked over two penalties either side of halftime to put France ahead in the 46th.
Richie Vernon pounced on a turnover ball to pop up a neat pass for winger Lee Jones to go over in the left corner, restoring Scotland's lead. But the home side couldn't hold on and the Scots have still only beaten France once since the Six Nations began in 2000.
Julien Malzieu trampled over Jones to feed Medard for Les Bleus' second try, which Parra again converted, before replacement flyhalf Lionel Beauxis knocked over a drop goal to push the lead to six.
Scotland couldn't recover and may again be facing Italy on the final weekend in a match to avoid the wooden spoon for last place.
