Television pictures showed flames leaping through barred windows of a single-storey wooden building.
The fire in a ward for bedridden patients was initially caused by an electrical fault in the building, a government statement said.
Thirty-one people were rescued during the incident, the General Prosecutor's office said.
Four people were taken to hospital, according to Azerbaijan's news agency APA.
The blaze, which broke out just before dawn, was fanned by strong winds. It took around three hours for fire fighters to extinguish it.
Azerbaijan's President Ilham Aliyev visited the site of the fire, the Republican Narcological Centre, after it was extinguished, local media said.
The former Soviet nation has a chronic shortage of rehabilitation places for its 30,000 registered drug addicts.
Nearly 3,000 people applied for places at the centre in 2015 alone, and a new clinic with 250 places has recently been built.Police and prosecutors said in a joint statement that the fire broke out at 6 a.m. local time (0200 GMT) on Friday in Baku and took three hours to extinguish.
An electrical short circuit is believed to have sparked the blaze at the clinic, which occupies an old, one-story wooden building.
Azerbaijan’s Health Ministry says 34 people have been rescued from the building and 200 patients and staff members were evacuated. Three people were hospitalized.
The office of the Azerbaijani president said he had arrived at the scene to oversee the rescue effort and ordered an investigation.
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