That beats the party’s previous record amid public dissatisfaction with President Francois Hollande’s Socialists.
The Socialists have retained control of Paris town hall with their candidate Anne Hidalgo wining 54.5% of votes, according to the polls.
If confirmed, the victory would be some consolation for Hollande’s ruling party, which is on track to lose its grip on dozens of major towns elsewhere across France.
Candidates backed by the National Front (FN) party looked set to secure the southern towns of Beziers and Frejus, the exit polls and partial tallies showed.
It was also likely to win the eastern town of Hayange, the outgoing Socialist mayor there said.
The party run by Marine Le Pen had already made a breakthrough in last week’s first round of voting by winning power in the northern town of Henin-Beaumont.
Its previous record was the three towns it won in 1995 local elections.
“Clearly we are entering a new phase, the duopoly of French politics has been broken and we must reckon with a third force,” Ms Le Pen said.
Socialists and mainstream conservatives have long dominated French politics…