The Red and Whites, who still remain in this season's Champions League courtesy of their aggregate victory over Arsenal earlier this week, took just five minutes to race ahead.
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Fabinho was on hand to strike home from a tight angle inside the area to put the visitors into an early lead, shortly before Anthony Martial coolly lofted the ball over the reach of Kossi Agassa on the 15-minute mark to double the advantage.
It could have been three prior to the interval had Martial kept his nerve, but he was unable to get the better of Agassa for a second time to ensure that Reims still had a glimmer of hope heading into the break.
France Under-21 international Martial was proving to be a key influence for Leonardo Jardim's side, with the first opening of the second period falling his way, but he could only graze the post having dribbled his way through on goal.
Despite the visitors' dominance, it was game on once more when Diego pulled one back with a fine strike for the home side, but Nabil Dirar was able to wrap things up by heading the ball beyond Agassa shortly after as Monaco secured a first win in Reims since 1978.
Defeat leaves Jean-Luc Vasseur's men in 15th place, just three points above the relegation zone.