Doussain converted a routine penalty in the 78th minute to set up an exciting finale against Ireland at the Stade de France next Saturday.
Ireland lead the table with six points, ahead of France on points difference before Sunday's clash between England and Wales.
The hosts, who have not beaten Les Bleus since a 20-16 victory in the championship in 2006, scored two tries through Stuart Hogg and Tommy Seymour with Greig Laidlaw adding the extras while Duncan Weir slotted home a penalty.
France, who had won 14 of their last 15 meetings with Scotland, had one Yoann Huget try to show for themselves with Maxime Machenaud's boot providing 11 points, to which Doussain added the winning three.
"We had a good start but some little mistakes cost us some points," Huget told French TV channel France 2.
"But we worked hard to come back into the game and we showed character."
"It was not easy. Maybe we were lacking confidence after the Wales game," said fullback Brice Dulin.
Philippe Saint-Andre's team, who were looking to bounce back from a 27-6 demolition by Wales, were terrible in the lineouts, unusually weak at the scrum and their paper-thin defence did little to help them.
Scotland were penalised for collapsing the scrum and Machenaud slotted home to give France a good start then benefited from the hosts' lack of discipline to add another penalty minutes later.
But Huget failed to control a high ball and Hogg, who had followed his own kick, touched it down and Laidlaw added the conversion to put the Scots ahead.
France, who finished with the wooden spoon last year, briefly regained control of the scoreboard thanks to Machenaud's boot, but Seymour benefited from a block on Jules Plisson to beat the defence and score Scotland's second try.
Laidlaw's conversion put them five points clear and Machenaud missed his first penalty kick on the stroke of haltfime with the Scots leading 14-9.
Huget, however, made up for his previous mistake when he intercepted a pass to make a 80-metre run for the line and with Machenaud's conversion, France took command early in the second half.
Saint-Andre, who has been tinkering with his halfback pairing since he took over from Marc Lievremont after the 2011 World Cup, replaced flyhalf Plisson with Remi Tales in the 48th minute.
Failing to roll away from a ruck, France were penalised shortly after the hour and Weir converted the penalty.
Les Bleus came close to scoring their second try shortly afterwards but Maxime Mermoz's knock-on ruined a well-crafted move.
Scotland, however, found a way to lose the game.
Two minutes from time, lock Tim Swinson did not roll away after a tackle and France were awarded a penalty, which Doussain, who had replaced Machenaud three minutes earlier, easily slotted between the posts.
- Sports & Recreation
- Yoann Huget
- Maxime Machenaud
- Greig Laidlaw
- Stade de France