The fatality and three of the injuries occurred when a rocket hit the mostly Christian neighbourhood of Bab Tuma in the Old City of Damascus, state news agency SANA reported, blaming "terrorists", the regime word for rebels.
Meanwhile, 27 people were wounded by regime mortar rounds fired into the northeastern rebel-held Qabun neighbourhood, the Syrian Observatory for Human Rights reported.
Some of them were seriously injured, said the Britain- based monitoring group.
Rebels frequently fire rockets on Damascus, which is largely in regime hands, from positions on the outskirts of the city.
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They often hit residential areas, causing dozens of deaths.
Government forces also regularly attack rebels entrenched in the Eastern Ghouta region east of the capital, which has been devastated by fighting since the outbreak of the Syrian conflict in March 2011.
More than 250,000 people have been killed in the war.