Two of Wales' three tries were donated by French mistakes, allowing Wales to pull off the biggest comeback win in the tournament since it expanded in 2000.
Wales ran its winning streak to 10, one short of the national record set more than a century ago.
"The biggest difference between the two sides is that we're a side who's forgotten how to lose, whereas on the other foot they're a side looking for confidence," Wales coach Warren Gatland said.
"It was strange. To win this tournament, you need a little bit of luck. You saw that last year with Johnny Sexton scoring a last-minute drop goal in a game France probably should have won (against Ireland)."
No. 8 Louis Picamoles barged over for an early try for France, and left winger Yoann Huget added a neat second. Neither was converted but flyhalf Camille Lopez added a penalty and dropped goal to make it 16-0 by halftime.
France ended a miserable 2018 with a home loss to Fiji. So French fans could scarcely believe the halftime score against the team second in last year's tournament and a strong candidate this year.
Wales had to hit back strongly and did so as scrumhalf Tomos Williams and North scored converted tries to pull within two of France just 11 minutes into the half. By then Wales had already two tries disallowed.
Wales went ahead for the first time with a penalty by replacement flyhalf Dan Biggar, Camille kicked France back in front, then North intercepted debutant Romain Ntamack's ponderous pass to Huget and sprinted home from 55 meters for the clinching converted score.
While Wales overcame a sloppy performance, France still seems chronically unable to make the most of a winning situation.
"It's not a question of mental strength," France coach Jacques Brunel said, somewhat surprisingly. "It's more a lack of control."
France's selection promised a more mobile, attacking approach than previous years. They made the ideal start as Picamoles bundled over in the right corner in the sixth minute after sustained pressure.
On a slippery night, Morgan Parra missed two difficult goalkicks for France, and Wales kicker Gareth Anscombe, who made all of his goalkicks in the 2018 championship, botched his first shot.
But a slick four-man passing move opened up the French defense, and fullback Liam Williams rode a tackle to get over the line. But the try was disallowed on video review by a knock on, which Williams could have avoided by passing inside to his left.
A physical French side unsettled Wales and a one-handed reverse offload by flanker Arthur Iturria released Huget into space on the left touchline and he scored in the corner.
Parra again missed the extras. Anscombe followed suit to make kickers 0-5.
Still, the Stade de France crowd had cause for optimism, having long been used to watching a brittle and gritty France side.
Since Les Tricolores reached the World Cup final in 2011, the side has finished no higher than third in the Six Nations — and that was only once. But the fans were enjoying the show and the players fed off their energy.
Lopez took over the goalkicking from his Clermont clubmate and nailed a penalty and long dropped goal on the buzzer to make it a dream first half.
The fans were lapping it up.
Then it all went wrong.
"It's frustrating, a feeling of deja vu, like South Africa (last year) and like many other times," France captain Guilhem Guirado said. "Every team in the world manages to control the second half. Not us, unfortunately."
Tomos Williams finished off a Josh Adams break for a Wales try on his tournament debut, then Huget spilled the ball on his tryline under no pressure, and North scooped it up and scored.
Anxiety spread in the home crowd as Biggar's penalty put Wales one point ahead with less than 20 minutes left.
Fear turned to relief when a mighty effort from the French scrum drew an error near the line, and Lopez clipped over the penalty for 19-17.
Then Ntamack undid the hard work moments later with a sloppy pass and watched North streak away.
Wales takes momentum to Italy next weekend. Guirado has to rally his crushed players for England at Twickenham.