The damage was discovered on Tuesday morning at a cemetery in the village of Quatzenheim, close to the border with Germany in the Alsace region, a statement from the regional security office said.
Photos show the Nazi symbols in blue paint spray-painted on the damaged graves, one of which bears the words "Elsassisches Schwarzen Wolfe" ("Black Alsacian Wolves), a separatist group with links to neo-Nazis in the 1970s.
The top security official for the region, Jean-Luc Marx, condemned "in the strongest possible terms this awful anti-Semitic act and sends his complete support to the Jewish community which has been targeted again," the statement added.
Mass rallies are planned in Paris and other French cities Tuesday to denounce a flare-up of anti-Semitic acts which culminated in a violent tirade against a prominent writer during "yellow vest" anti-government protests last weekend.
Political leaders called the rallies after a protester was caught on video calling the philosopher Alain Finkielkraut a "dirty Zionist" and telling him that "France belongs to us."
Last year, French police recorded a 74 percent surge in reported anti-Jewish offences, causing alarm in a country that is home to the biggest Jewish population in Europe.
"It just doesn't stop, it's shock after shock," Maurice Dahan, the regional head of France's main Jewish institution, the Israelite Central Consistory of France, told AFP after the attack in Alsace.
"I don't know how long we are going to carry on... It makes me feel sick.