Martial Godo nodded Strasbourg ahead after seven minutes. Diego Moreira scampered down the right wing, chipped to the far post for Thilo Kehrer who headed back into the six yard box for the diving Godo.
Julio Enciso started a flurry of goals when he doubled the lead after 55 minutes, beating the offside trap and skipping past goalkeeper Philipp Koehn before slotting home.
Mika Biereth replied seconds after the restart, guiding a shot inside the far post.
In the next three minutes, Enciso had a shot blocked, chipped over the bar and then added a second, pouncing on a loose ball to beat the diving Kohn.
Monaco, who had pulled out of a recent swoon by squeezing into the Champions League knockout play-offs and beating Rennes 4-0 in Ligue 1, subsided to another defeat.
Meanwhile Gary O'Neil won for the fourth time in five games since taking over as Strasbourg coach after Liam Rosenior moved to Chelsea
The match was played against a background of fan discontent.
Visiting Monaco fans held up a banner reading "management resign".
Just before kick-off, Strasbourg fan groups published an open letter to the governing bodies of French football and to the French Minister of Sport, denouncing their team's "subservience" to Chelsea - the flagship of BlueCo, which owns both clubs.
In the quarter-finals scheduled for early March, Strasbourg will face the only second-division club left in the competition Reims.
The highlight of the draw, made on Thursday, is the match between Lyon, unbeaten in 11 games after their victory over Laval in the round of 16 on Wednesday, and Lens, second in Ligue 1.
In the other two ties, Marseille will host Toulouse and Lorient will entertain Nice.