Luca Cari, spokesman for the national firefighters' service, confirmed the death toll to Reuters, saying that the fire had been put out and that its causes were under investigation.
He said three people were hospitalised in a critical condition, 15 with serious but not life-threatening injuries and about 50 with light injuries.
Milan Mayor Giuseppe Sala, speaking to reporters on the scene, said the fire was extinguished very quickly but produced fumes that intoxicated a lot of residents.
Italian news agencies, citing regional emergency services, had previously reported that 81 people had been hospitalised, including two in a critical state.
According to media reports, the fire broke out at 1120 GMT on Thursday in a facility in the south-eastern Corvetto neighbourhood.
(Reporting by Alvise Armellini, editing by Giselda Vagnoni)
