Mr Assad's forces also came under attack in the east, where a suicide car bomb struck a military intelligence compound in the city of Palmyra, causing dozens of casualties, they said.
Authorities in Damascus closed the main Abbasid Square and the Fares al Khoury thoroughfare, as fighters attacked road-blocks and fortifications with rocket-propelled grenades and mortars.
"The areas of Jobar, Zamalka, al Zablatani and parts of Qaboun and the ring road have become a battleground," activist Fida Mohammad said from the district of Qaboun.
Another activist said an army tank at the main al Kabbas roadblock on the ring road had been des-troyed. Residents reported explosions to the east and north of the capital.
In Jobar, a working-class Sunni Muslim area adjacent to Abbasid Square, mosque speakers chanted "God is Greatest" in support of fighters who attacked roadblocks, activists said.
They said tanks just outside the walls of Old Damascus, shelled southern districts of the city.
Syrian state television said: "Our noble army is continuing its operations against the terrorists in Irbeen, Zamalka and Harasta and Sbeineg, destroying the criminal lairs."
Mr Assad's symbols of power came under attack in Palmyra, 140 miles northeast of Damascus, on the main road to the oil-producing east of the country.
A bomb destroyed part of the back wall of the military intelligence compound near the Roman-era ruins then a suicide car bomber drove through, detonating the vehicle, activists said.