Monaco, second in Ligue 1, were behind after a minute against third-tier Vannes before Radamel Falcao equalised two minutes later.
Yannick Aguemon restored the hosts' lead, only for James Rodriguez to again level the scores before the break, with Yannick Ferreira Carrasco ending any chances of an upset four minutes from time to book Claudio Ranieri's side a place in the last 32.
Lille also found themselves behind early on when they travelled to fifth division Amiens AC, but fought back as a Salomon Kalou brace helped them to a 3-1 win at the Stade Jean Bouin.
David Bellion was the hero for Francis Gillot's Bordeaux at fourth-tier Raon-l'Etape, as he struck four minutes from time to help the holders to a 2-1 win.
Nice came out on top against top-flight rivals Nantes thanks to goals from Didier Digard and Christian Bruls, while Bastia sent last year's finalists Evian out courtesy of Florian Raspentino's 20th-minute effort.
Ten-time winners Marseille needed extra time to see off Reims, with Andre-Pierre Gignac finding the net for a fourth successive game to score twice and settle the tie, while Lyon thrashed La Suze 6-1 thanks to braces from Bafetimbi Gomis and Alexandre Lacazette.
Auxerre also struck six, seeing off Marcq-en-Baroeul 6-2 having been three goals ahead after nine minutes, while Iris Club de Croix needed penalties to see off Saint-Amand after the scores were level at 1-1 after 120 minutes.
FA Ile Rousse Monticello and 2009 champions Guingamp progressed with 2-0 wins over Cayolle and Bourg-Peronnas respectively, with Ligue 1's bottom side Ajaccio making the most of their break from league football.
Ajaccio came from behind to edge Quimperle 2-1 and M'Baye Niang scored from the spot on his debut to help Montpellier see off Rodez 2-0.
Sete enjoyed a 2-0 win over Carquefou, while Paris Saint-Germain against Brest and Saint-Etienne's visit to Cannes were both postponed due to bad weather.
