Defeat means France, who were beaten 19-10 by Wales a fortnight ago, remain on four points and cannot overhaul Eddie Jones's side when they meet in Paris next weekend, with England looking for a grand slam.
England, who secured a thrilling 25-21 victory over Wales on Sunday morning (NZT), have eight points after four matches and secured the title for only the second time since their 2003 World Cup triumph.
France opened the scoring in the fourth minute with a try by hooker Guilhem Guirado, but Stuart Hogg and Duncan Taylor scored two tries in the space of three minutes at the end of the first half.
Halfback Greig Laidlaw converted the second try and added three penalties in the second half for a personal haul of 11 points.
Gael Fickou ran in France's second try on the stroke of halftime, but Tim Visser's third for Scotland in the 67th minute saw the hosts leapfrog France into third place on four points.
"We've had poor performances in the past," fullback Hogg said to the BBC. "But to make amends today is brilliant.
"I can see how much it means to our home fans, who were unbelievable."
"This is the reason why we pull on this jersey and make so many sacrifices," said Laidlaw. "We make so many sacrifices, and that was for the fans.
"To lead this team at Murrayfield is something that will stay with me forever. We were all as one today. It was outstanding."
Scotland ended a two-year wait for a Six Nations victory with a 36-20 triumph over Italy in Rome in February ending a run of nine successive losses in the tournament.
They started poorly and soon fell behind when France captain Guirado barrelled over the line.
The home side lost first five-eighth Finn Russell to injury in the aftermath of the French try, but his replacement Peter Horne was influential as the Scots scored their first try through Hogg, building on the fine form he demonstrated in Rome.
Taylor added the second try two minutes later in the outstanding moment of the first half. Taking a penalty in the Scottish half, the centre decided to go the distance, dashing down the right flank to score with a magnificent solo effort.
France responded immediately as Fickou bundled over the line following sustained pressure in the 40th minute to go in 18-12 behind at halftime and set up a tense second period.
The visitors kept their intensity up after the break, with Hogg's penalty cancelled out by halfback Maxime Machenaud's pair of kicks.
The tiring Scots extended their advantage in the 67th minute against the run of play when wing Visser scored in the corner following fine interplay from Hogg and Laidlaw.
"We played a weak game," complained France coach Guy Noves to France 2. "It's more us losing the game than them winning it."
"We made stupid mistakes," said Machenaud.
'Fantastic achievement'
England coach Eddie Jones hailed a "fantastic achievement" by his team.
"It's a fantastic achievement," Jones told BBC Sport.
"We've made sure the players are responsible, we play in a style that suits our resources and the players enjoy themselves."
England were knocked out of the World Cup on home soil last year after losing at Twickenham to Wales and Australia in the group stage.
Coach Stuart Lancaster was sacked after the tournament and Jones appointed as England's first foreign coach in November.
The Australian paid particular tribute to his blindside flanker Chris Robshaw, the team captain under Lancaster who was widely criticised following the World Cup debacle.
"To be splashed on the papers, basically the villain of English rugby, and have to endure the captaincy debate, which I didn't help but was necessary," Jones said.
"To come back and play such a pivotal role in the resurrection of the team, I think it is just outstanding," Jones added.
"It would have been easy for him to go back to Harlequins and ply his trade there and have a nice easy life but he really wanted to continue playing for England."
Robshaw has made the adjustments himself, Jones said.
"The guidance I've given him is what can his new goals be - he can be the best No 6 in European rugby," the Australian added.
"If you were picking a Six Nations team now, he would definitely get in a best 15 at No 6."
