While Saturday was a day of upsets with fifth division amateurs Chambery knocking out Ligue 1 Brest and Angers (Ligue 2) eliminating Girondins Bordeaux, the elite teams generally enjoyed better fortunes.
French international defender Clerc, who joined Nice last year, fired home from close range in the sixth minute of extra-time to give Nice a deserved win over Lyon.
The hosts dominated possession but lacked penetration while Lyon missed two clear chances through Kim Kallstrom.
Lyon finished the game with 10 men after defender Dejan Lovren was sent off in extra-time for a second bookable offence.
Tulio de Melo's goal in the 40th minute was enough for Ligue 1 leaders Lille to beat neighbours Wasquehal of the fifth division 1-0.
Stade Rennes initially struggled at sixth-tier Vaulx en Velin but prevailed 2-0 with second-half goals by midfielders Alexander Tettey and Yacine Brahimi.
Holders Paris Saint-Germain looked on course for a routine win at fifth division Agen when Mathieu Bodmer opened the scoring in the 13th minute but the amateurs did not give up without a fight.
Mamoudou Daffe headed home four minutes before the interval to level the tie but Nene was at his brilliant best after the break, setting up Peguy Luyindula and Guillaume Hoarau as PSG opened up a two-goal lead.
Anthony Vandersnick pulled one back for Agen to make the final score 3-2.
All seven Ligue 1 teams left in the competition avoided each other in the draw for the last 16 with matches due to be played on February 1/2.
