The rockets hit Al-Dabbeet hospital in the Muhafaza district, state news agency SANA said.
Rebel rocket fire on other government-held neighbourhoods killed another 11 people, it added.
The Syrian Observatory for Human Rights said rebels had fired rockets and shells on government-controlled western districts of the city throughout the day.
After a lull through the morning, regime air strikes on rebel-held eastern areas resumed in the afternoon, an AFP correspondent reported.
Moscow has faced mounting pressure from Washington to rein in air raids by its Damascus ally after a hospital and three clinics were hit.
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