Former England forward Cole scored in the 22nd, 59th and 65th minutes as the French champions cantered through with Eden Hazard, who converted a penalty, and Ireneusz Jelen, who hit a double, also on the scoresheet.
Olympique Marseille's visit to the Stade de France to face third division Red Star ended in a 5-0 win thanks to a Jordan Ayew double with his older brother Andre, Mathieu Valbuena and Benoit Cheyrou also on target.
Stade Rennes also went through thanks to a 3-0 home victory against fellow Ligue 1 side Nancy and Girondins Bordeaux reached the next round by winning 4-2 on penalties at Saint-Etienne after the game ended in a 1-1 draw after extra-time.
However, Lorient, Stade Brest, Sochaux and Caen all lost to lower level opposition.
Lorient slumped to a dramatic 4-3 defeat at second-tier Le Havre with Yohann Riviere netting the winner 10 minutes from time.
Brest went down 2-0 at third-division Niort to goals by Arnaud Gonzalez and Jimmy Roye either side of the interval, while Sochaux were thrashed 4-1 by Ligue 2 Bastia.
Caen lost 4-2 at home to Ligue 2 Troyes after extra-time.
Top-flight AJ Auxerre were pushed to extra-time before winning 1-0 at fifth-division Chambly courtesy of a late Dariusz Dudka penalty.
Amateurs AS Marck, who also play in the fifth tier, lost 2-0 at home to Ligue 1 Nice adding to the gloom affecting the area with the local coach and several players battling to keep their jobs at cross-Channel ferry operator SeaFrance.
But another fifth division team Limoges, best known in France for their basketball team's European Cup victory in 1993, fared much better by beating Ligue 2 visitors Boulogne 1-0.
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