The rebel attack came two days after they threatened to retaliate for deadly air raids by the Syrian regime against an opposition-held area on the edge of the capital.
"Five civilians and two soldiers have been killed and dozens more wounded, as rebels in the Eastern Ghouta area fired more than 43 locally made rockets and mortar rounds at several areas of central Damascus," the Britain-based Syrian Observatory for Human Rights said.
Among the areas hit were the Al-Maliki and Mazzeh neighbourhoods, as well as Arnus and Sabaa Bahrat squares, said the Britain-based group.
AFP journalists near Sabaa Bahrat square heard the blasts, while ambulances and fire trucks were seen rushing to the area.
State news agency SANA said the army fired back at the source of the incoming fire in Eastern Ghouta, without providing any details of casualties or naming the areas hit.
The attack came two days after Zahran Alloush, the head of the rebel Jaysh al-Islam (Army of Islam) group, warned on Twitter that his forces would launch a "rocket campaign against the capital" from Sunday.
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He wrote that rockets would "rain down every day... in retaliation for the regime's savage air raids" against residents in Eastern Ghouta.
Government aircraft on Friday carried out a series of deadly raids against rebel-held Hammuriyeh in the besieged Eastern Ghouta area east of Damascus.
The Observatory said 56 people were killed, among them six children. Only five of the dead were fighters, said the group, which relies on a network of activists and medics on the ground.
Alloush's Jaysh al-Islam is the most powerful rebel group in Eastern Ghouta.
Tens of thousands of civilians trapped in the area suffer extreme shortages of food and medicine, activists say.
Syria's conflict began as a peaceful revolt demanding democratic change, but evolved into a brutal civil war after President Bashar al-Assad's regime unleashed a massive crackdown against dissent.
More than 200,000 people have been killed since March 2011 and half the population has been forced to flee their homes.