PARIS—The National Front’s victory in a first round of regional elections left France’s mainstream parties struggling to define a strategy to keep Marine Le Pen’s anti-immigration party from winning control of several of the country’s regions.

The week heading into a second round of voting is traditionally a time for France’s mainstream political parties—President François Hollande’s Socialists and former President Nicolas Sarkozy’s center-right forces, now known as Les Républicains—to brush off smaller third-party challengers...