Big-spenders Monaco were given a scare in the French Cup on Sunday before a late Yannick Ferreira Carrasco winner eased them past third division Vannes 3-2 and into the last-32.
Top flight leaders Paris Saint-Germain had their match at Brest postponed until Tuesday at the weather-hit Brittany club, while Marseille, Lille, Bordeaux, Lyon, Guingamp, Rennes, Montpellier and Ajaccio all advanced.
Ligue 1 clubs Nantes, beaten 2-0 by Nice, and Evian, downed 1-0 by Bastia, were knocked out by top flight rivals.
"It was a difficult match against a very good Nantes team," said Nice midfielder Eric Bautheac after their 2-0 win.
"We finished 2013 well and have started 2014 well which is good because it's the end of our bad run of results. We've started the year in perfect fashion and have put our problems behind us," he added.
Monaco's Colombian striker Radamel Falcao cancelled out a first minute strike from Hendrick Cakin 60 seconds later, only for the hosts to move ahead again nine minutes before half-time through 21-year-old French striker Yannick Aguemon.
Falcao's compatriot James Rodriguez tied the match just before the break before Belgian midfielder Ferreira Carrasco avoided extra-time and a potentially embarrassing exit, with the winning goal for the Ligue 1 high-flyers four minutes from the conclusion of a spirited Cup tie.
Monaco brought a squad assembled for a budget of 130 million euros to the west of France for a fixture against a Vannes outfit put together for just three million euros.
In the final match of the weekend, Marseille needed extra-time goals from Pierre-Andre Gignac to see off the challenge of Reims 2-0 at the Velodrome.
A tight match on the south coast was broken open by 94th and 115th minute strikes by Gignac.
Elsewhere, holders Bordeaux ground down third division Raon l'Etape 2-1, Lille came back to down Amiens 3-1, helped by a brace from Ivory Coast striker Salomon Kalou, and Montpellier defeated lower tier club Rodez 2-0.
Seven-time French champions Lyon thrashed La Suze 6-1, where Christian Grenier, Alexandre Lacazette and Bafetimbi Gomis each scored twice.
On Saturday, third division Yzeure caused a shock by knocking out Lorient 1-0.