Wales claimed their second Six Nations title in three years after France lost 27-23 to Scotland in the last game of the championship as the visitors celebrated their first win in Paris since 1999 on Friday.
France looked set to win their fourth match when they won the ball in the last play in stoppage time, only for Brice Dulin, who had scored the decisive try in last Saturday's dramatic 32-30 victory against Wales, to run with it instead of kicking it out in an ill-advised attempt to secure a fourth-try bonus point.
Les Bleus, who scored tries through Dulin, Damian Penaud and Swan Rebbadj, lost it again and Scotland went on to score a last-gap try through Duhan van Der Merwe to seal victory.
The result still left Scotland in fourth place despite beating England at Twickenham and France in Paris, just behind Ireland on points difference with 15 points.
Wales have 20 points from four wins with France, who needed a win by 21 points and with a bonus point to leapfrog them for their first title since 2010, second on 16 points.
England are fifth on 10 points with Italy taking the wooden spoon having lost all their games without a bonus.
Scotland needed to win by eight points to get their highest finish in the championship since they won the last Five Nations in 1999.
They came up short, but left the pitch with pride after Van der Merwe's two tries, another one by David Cherry and points from the boot of Finn Russell and Adam Hastings.
With the game being played outside the international window, coach Greg Townsend could select only five players from English clubs, meaning that back-three Sean Maitland and prop Jamie Bhatti missed out.
Both teams resorted to kicks early on and France were the first to score through Romain Ntamack's penalty.
But they found themselves facing a higher mountain to climb when Scotland moved 7-3 ahead as Duhan van der Merwe picked the ball from a ruck and touched down, with Russell converting.
Russell added another penalty.
Les Bleus made basic errors and could only see Scotland's try line from a distance, until Dulin dived over after being found on the inside by Damian Penaud as France were rewarded for a long domination spell.
Scotland paid for their ill discipline with a Stuart Hogg yellow card on the stroke of half-time.
After a first foiled attempt, France grabbed the second try when Penaud grabbed Virimi Vakatawa's brilliant offload before chipping it over and touching down to extend the lead to eight points.
But the visitors went ahead again with a penalty and a try by Cherry, who leapt over from a maul before Russell converted.
France hit back when Rebbadj went over after being set up by Antoine Dupont close to the line.
Russell was then sent off with nine minutes left for elbowing Penaud in the throat, but France lost their advantage as replacement scrum-half Baptiste Serin was sin-binned three minutes later.
It seemed over for Scotland but Dulin lost the plot in stoppage time for a memorable error that gave Scotland a last chance, which they made a feast of as Van der Merwe managed to dive over in the corner.
"I'm so proud of the team," said coach Gregor Townsend, a try scorer in Scotland's last Paris win 22 years ago. "We've come here with a bit of adversity, not a full squad, got an injury on Wednesday, got a red card, yellow card and we've had to come back against a very very good side. They showed courage, togetherness, determination and skill to win. So it was a great end to a really promising season for us."
Regarding the last gasp assault, he said: "It took a lot of knocking on the door, which is credit for France's defence, but the players found a way. I'm so proud of them and even though we've finished fourth it feels like one of our best-ever seasons.
"Over the last two years we've been competitive in every game and the players will take confidence from that."