The five lived in a flat on the fifth floor of the six-storey building in a poor area of Gennevilliers, northwest of Paris, they said, adding that the cause of the fire remained unknown.
Around 100 firefighters fought the blaze that began in a fourth-floor flat, rescue workers said.
Fifteen people were taken to hospital mostly suffering from smoke inhalation, while one firefighter suffered hand burns and another from heat exposure.
Interior Minister Manuel Valls visited the scene, as did the mayor of Gennevilliers, who spoke of "the death of an entire family".
Firefighters were reluctant to confirm that the five dead were from the same family until formal identification was carried out. But the building caretaker confirmed that a family of five lived in the flat that was devastated.
The dead included a 12-year-old, a 15-year-old and an 18-year-old, officials said.