The sweep around dawn in Strasbourg came weeks after authorities announced a new push to prevent French militants from travelling to Syria.
Interior Minister Bernard Cazeneuve said the six were suspected of participating in a jihadist network, and recently travelled to Syria.
In a brief statement at his ministry, Cazeneuve said the sweep "went down perfectly" and showed "the total determination of the government to fight with all its power against terrorism and the enlisting of youths in violent radicalisation." The suspects were in police custody.
Hundreds of young French militants have travelled to Syria. Authorities fear they could return home with fighting skills and ambitions to carry out attacks.