The 10-time winners took the lead after just three minutes through striker Andre-Pierre Gignac, only for goals from Alexy Bosetti and Neal Maupay to turn things around for the visitors.
Winger Florian Thauvin restored parity after 24 minutes, but a first-half injury-time own goal from defender Souleymane Diawara and a 51st minute Christian Bruls penalty put Nice in command.
Gignac's second gave Marseille hope just before the hour-mark, but Fabrice Abriel's 88th minute effort effectively killed the game off, despite a stoppage-time goal at the right end from Diawara.
The 2011 winners Lille remain in the hunt to lift the trophy again following a 3-0 win at fifth-tier side Iris Club de Croix, with Simon Kjaer, Tulio de Melo and Divock Origi netting in a comfortable victory for Rene Girard's men.
Anders Konradsen and Silvio Romero scored as Rennes beat third-tier side Boulonge 2-0. However Bastia, Ajaccio and Sochaux all suffered surprise defeats.
An extra-time strike from Deme N'Diaye gave Ligue 2 promotion-hopefuls Lens a 2-1 success over Bastia, whose Corscian rivals Ajaccio were beaten 2-0 at home by second-tier Caen.
Sochaux succumbed to a 1-0 loss at Angers, the Brittany club - who sit second in Ligue 2 - progressing thanks to a second-half effort from Djibril Konate.
Claudio Beauvue struck the winner as Guingamp needed extra-time to beat fourth-tier Concarneau 3-2, while CA Bastia overcame Niort 3-2 on penalties in a tie that finished 2-2 after extra-time and both sides were reduced to 10 men.
Sete also progressed as they beat Jura Sud 2-0.
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