Holders Paris St Germain avoided any such drama as they eased through with a 4-0 win at Stade Rennais courtesy of a Julian Draxler double.
In Chambly, goals from Guido Carrillo, Thomas Lemar and Kilian Mbappe put Monaco 3-0 ahead.
But the hosts fought back after the break as a Gregory Gendrey goal and a double by Anthony Soubervie, who netted the equaliser in stoppage time, levelled the tie.
Monaco defender Andrea Raggi was shown a straight red card for a dangerous tackle in the 89th minute, four minutes before Soubervie's second goal.
Monaco kept their wits and went 5-3 up in extra time thanks to goals from Kevin NDoram and Kamik Glik.
Romain Padovani pulled one back for Chambly but despite being down to 10 men, Monaco held firm.
(Reporting by Julien Pretot, editing by Pritha Sarkar)
Share
