Firefighters have extinguished the blaze at Dovatora Street building 11/2, two blocks away from the Luzhniki stadium, near Sportivnaya metro station, the spokesman for the Moscow department of the emergencies ministry said.
"According to the latest information, a child died, and three other residents of the building, including another child, were injured. Rescuers have evacuated 15 people from the burning building," he said.
"The fire was extinguished at 17:30 [13:30 GMT]. Investigators from the state fire safety agency are establishing the cause of the fire."
Preliminary reports said the fire started on the ground floor of the building. Witnesses earlier said they could hear people trapped in the building calling for help.
A spokesman for the Moscow department of the Emergency Situation Ministry earlier said that all residents had been evacuated from the building.