Elsewhere, a car bomb killed at least 19 security officers.
The clashes in Damascus were the most intense violence to hit the capital in weeks. The sounds of gunfire and blasts from government shelling of rebel-held areas could be heard in most neighborhoods. Activist videos online showed shell explosions dotting rebel areas east of the city, covering them with clouds of smoke.
Government troops blocked traffic at a key intersection in the city's northeast and on a number of nearby roads.
Damascus has not experienced the same fierce fighting as other Syrian cities like Aleppo or Homs, where whole neighborhoods have been destroyed. While the government has lost control of parts of those cities, it has kept a tight grip on the capital, despite rebel attempts to storm the city center from enclaves on its outskirts.
Much of Wednesday's fighting centered on the northeastern neighborhood of Jobar, which is bisected by the Damascus ring road. Rebels, who control the area east of the road, launched attacks on army checkpoints in the regime-controlled western part in a push to seize the road, one of the capital's most important thoroughfares.
Rami Abdul-Rahman, director of the Britain-based Syrian Observatory for Human Rights, said Wednesday's shelling of Jobar and Qaboun is part of a wider government offensive against towns and villages near the capital that have been opposition strongholds since the beginning of the uprising against President Bashar Assad in March 2011.
A government official said army troops are pursuing rebels in the suburbs of Harasta, Sbeineh, and Jober. The official spoke on condition of anonymity because he was not authorized to brief reporters.
The Observatory also reported two car bomb blasts in the ancient city of Palmyra in central Syria. One targeted the local branch of Syria's military intelligence agency, killing at least 19 security officers and wounding many more. A second and apparently coordinated bomb hit another security office. Eight civilians were wounded in gunfire and explosions following the two attacks, the Observatory said.
Syria's state news agency confirmed the blasts but said two suicide bombers blew up their cars near a garage, killing and wounded an undisclosed number of people.
No one claimed responsibility for the blasts. The government blamed them on "terrorists" - its blanket term for the opposition.
Homs has been an opposition stronghold throughout Syrian uprising. The province and its capital of the same name were the scene of mass protests early in the revolt, which has since developed into a civil war that has turned urban centers like Homs and the northern city of Aleppo into battlefields.
The United Nations say more than 60,000 people have been killed since the conflict started. At least 700,000 Syrians have fled, seeking shelter in neighboring countries like Jordan, Lebanon and Turkey. More than one million people have been displaced within Syria during 22 months of fighting, according to aid agencies.