Artillery shells and rockets landed in the heart of the capital as part of a surprise rebel attack launched in the Jobar district.
The assault began early on Sunday with car bombs and suicide attacks, activists said. State media said secret tunnels were also used.
The military says it managed to push the rebels back.
Government warplanes launched more than 30 air strikes on rebel positions, according to the Syrian Observatory for Human Rights, a group of activists monitoring the conflict.
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Damascus only has a few opposition-held areas, and Jobar is the closest to the city centre. Control of the war-damaged area - which is split between rebels and jihadists on one side and government forces on the other - has been fought over for more than two years.
AFP correspondents in Damascus said the army had closed routes into the strategically important Abbasid Square as explosions reverberated across the city.
The Observatory said the rebels had initiated the attack to relieve pressure on fighters under attack from government forces in the districts of Barzeh, Tishreen and Qabun.
Last Wednesday, at least 31 people  were killed in a suicide bomb attack at the main court complex in the centre of Damascus.
Later, another suicide bomber attacked a restaurant in the western district of Rabweh, injuring more than 20 people.
The attacks came on the sixth anniversary of the uprising against President Bashar al-Assad.
Syrian security forces have engaged in fierce clashes with rebels on the eastern outskirts of Damascus.