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Nice found themselves slowly growing into the game, but they came unstuck just after the half-hour mark when goalkeeper Hassen was given a straight red card for bringing down Braithwaite inside the area.
The dismissal gave Braithwaite the chance to convert from the spot, which he willingly completed with a neat finish beyond substitute stopper Yoan Cardinale, giving Toulouse a one-goal lead at the break.
The home side were able to make their numerical advantage count after half time, when Ben Yedder added his side's second goal of the night having converted into the bottom corner from the centre of the box.
Nice might have made a fight of things late on, but Hatem Ben Arfa was unable to find the target from the centre of the box, leaving the hosts to claim an important victory down at the foot of the table.