Ten people were also injured, some of them seriously, and more than 100 people were evacuated.
The alarm came in at 1:42 a.m.local time. Officials say fire crews arrived in less than 10 minutes -- and took only 20 minutes to extinguish the fire.
But it was still too long, as relatives of the dead wept today outside rehabilitation clinic No. 17 in southern Moscow.
Yury Nenashev, the Russian Emergency Situations Ministry's top fire official, told reporters on the scene that he's already "90 percent" certain the blaze was deliberately started.
"Now we can already say for sure that the fire started in the canteen area. We know where the fire started. And we are 90 percent certain that it was arson," Nenashev said.
However, Moscow city prosecutor Yury Syomin said all possible causes of the fire are still being checked.
"We will be investigating all possible causes of the fire, including the possibility that it was deliberately started. Someone could have started the blaze on purpose," Syomin said.
Many of the victims -- mostly patients undergoing treatment for drug and alcohol addiction -- are reported to have died because of smoke inhalation.
Nenashev said attempts by victims to flee the blaze were hampered by a locked gate and metal bars over windows.
"The second escape stairway was covered by heavy smoke, so people could not escape, and they died. Four hospital staff members who were at the scene could not help the patients to escape, and were forced to flee the scene themselves," Nenashev said.
The Emergency Situations Ministry said it had tried to close the hospital earlier this year because of fire safety violations. These violations included the metal bars on the windows and staircases. A court turned down the request.
Russia has seen several fatal fires at hospitals where drug addicts or mentally ill people are treated. They are often in old, neglected buildings, where patients are held in secure conditions.
In December last year, seven people died when a fire broke out overnight at a hospital near Moscow treating people for nervous disorders. In March 2005, seven people were killed in a fire at a drug treatment center in Samara, in central Russia.