Gregor Townsend's side dominated large spells of an open, fast-paced contest, eventually outscoring France seven tries to six. Darcy Graham and Kyle Steyn both grabbed two tries, with Pierre Schoeman, Ben White and Tom Jordan also crossing the line. Finn Russell added 15 points with six conversions and a penalty.
Scotland struck early through Graham before France briefly fought back to lead, but the hosts regained control with Steyn's first try to take a 19-14 lead into half-time.
The Scots pulled clear after the break. White scored soon after the restart for the bonus-point try, Steyn intercepted to add his second, and Graham finished another superb move as Scotland surged into a commanding lead.
France rallied late with tries from Antoine Dupont and Thomas Ramos to secure a bonus point, but Russell's late penalty ensured Scotland finished deserved winners in a high-scoring Murrayfield classic.
Scotland captain Sione Tuipulotu told BBC One: "Considering how this tournament started for us, we believe now, the tournament is not over for us.
"I could not be prouder to be captain of this team. We've got ourselves an opportuntiy next week.
"We stuck together after a tough first round and tough autumn and have now given ourselves one more job to do next week.
"Couldn't be happier for him [Gregor Townsend], we rallied behind our coach. One more week to finish the job."
Player of the match Kyle Steyn confirmed the injury that forced him off isn't too concerning.
He told BBC One: "Just a big ugly gash, yeah [I'll be back for next week],
"I think we knew you couldn't come and try contain them, you had to fire shots at them.
"You just knew it was going to be a great day. It was all about making sure we fired the first shot.
"I was more worried about how much we were celebrating, we needed to get our feet on the ground and keep going."
Matt Fagerson played a big part in a huge collective effort from the home pack. Speaking To Scottish Rugby after the match, Fagerson said: "Yeah, pretty ecstatic. It's probably the one of the best atmospheres I think there has ever been at Murrayfield and that's something that I've been a part of. Yeah, it was electric from the first half through to the second and yeah, I mean, I can't thank the fans enough.
"Obviously we come out the box again like we talked about a fast start and we did that in the first half so it's kind of emulating in the second. I thought we did that really, really well and then a couple injuries to a few of our key boys and yeah, we maybe lost our way a little bit in the last 10, but yeah, the effort was incredible."