The blaze happened early on Wednesday. The five victims shared a flat on the fourth floor of the seven-storey block.
Eighteen people, including an infant, were hurt and most are now in hospital. About 100 firefighters tackled the fire, using a ladder to rescue people.
A firefighter quoted by French TV said the building was in good condition and it is not clear how the fire started.
Two firefighters who suffered burns to their hands were among the 18 injured.
The fire broke out shortly after 04:00 local time and was extinguished at about 06:00. The firefighters managed to prevent it spreading widely through the building.
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