Willy Sagnol's side, whose three victories before tonight have all come at home, started brightly as Wahbi Khazri's shot was blocked before Enzo Crivelli was also crowded out as he tried to take aim at goal.
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Bordeaux hit back on 36 minutes as a fast counter-attack from the hosts culminated in Nicolas Maurice-Belay's finish from only a few yards out.
Just after the restart, Sagnol's men made it 3-1 after Jaroslav Plasil got on the end of Crivelli's cross before slotting low past Subasic.
The home side could have scored a fourth when Khazri's menacing delivery was headed against the crossbar by Crivelli as Bordeaux threatened to run riot.
Monaco tried to fight back in the dying stages, with Almamy Toure heading narrowly wide, but Bordeaux managed to hold on for only their fourth victory in 13 league matches this season.