Scotland believed they had crossed for what would have been the match-winning try with the clock in the red, but Australian referee Nic Berry did not agree.
The Television Match Official took several minutes to review the footage and could not find conclusive evidence to overturn that decision, even though it looked as though the ball may have touched the turf.
It was initially grounded onto the boot of a French player, but then appeared to slip down onto the grass.
The win will be a mighty relief for France coach Fabien Galthie, who has come under enormous pressure following their dismal opening 38-17 loss to Ireland in Marseille.
Trailing by six points with 11 minutes remaining, Bielle-Biarrey had three defenders around him but a clever chip and electric pace he gathered to dot down and fullback Thomas Ramos’ conversion gave the visitors the lead for the first time.
Ramos kicked a further penalty as La Marseillaise rang around the ground, with centre Gael Fickou scoring his team’s other try in the first half.
Scotland dominated much of the match and led 13-10 at halftime thanks to scrum-half Ben White’s score, but could not turn pressure into points on several occasions, and were made to pay in a game they looked like winning for 69 minutes.
France struggled in the lineout and were sloppy with the ball in hand in slippery conditions, while the breakdown was a lottery and not controlled by the match officials.
White crossed for Scotland’s only try after a sweeping move down the right wing involving debutant Harry Paterson and centre Huw Jones. They made the yards before White was fed on the inside and the scrumhalf fought off two tacklers to dot down.
Scotland were down to 14 players while flanker Matt Fagerson received treatment and France were able to use the width of the field for Fickou to canter in for their first score.
The visitors were reduced to 14 players just before halftime when prop Uini Antonio produced a no-arms tackle and was sent to the sin-bin.
It was the first time France have trailed at halftime in back-to-back Six Nations matches since 2016.
The visitors were dealt a blow when their outstanding captain Gregory Alldritt left the field on a stretcher with a nasty gash on his leg on 50 minutes, but through Bielle-Biarrey and perhaps a little luck they will leave Edinburgh with a win.
