Troops were also said to have deployed in the western suburb of Mezzeh.
Fighting also continued for a third day in Aleppo, where activists said a building had collapsed under tankfire.
The violence follows a week in which rebels made major advances, taking control of several parts of Damascus, seizing border crossings and claiming an attack that killed four top security officials, including the defence minister and President Bashar al-Assad's brother-in-law.
Meanwhile, the UK-based Syrian Observatory for Human Rights has reported that at least 19,106 people had been killed since March 2011.
They included 13,296 civilians, including those who had taken up arms, as well as 4,861 security personnel and 949 army defectors, it said.
The UN said in May that at least 10,000 people had been killed, while in June the Syrian government reported that 6,947 Syrians had died, including at least 3,211 civilians and 2,566 security forces personnel.
Firepower The BBC's Jim Muir in Damascus says government forces seem determined to drive the rebel Free Syrian Army (FSA) completely out of Damascus and are setting about it quite systematically.
Having regained one district close to the centre of the capital and another on its north-eastern edge, troops are now focusing on Mezzeh and Barzeh, our correspondent adds.
The attack on Barzeh by the army's fourth division, which is commanded by the president's brother Maher, had sent residents fleeing from the area, the Syrian Observatory reported.
Helicopter gunships were firing rockets and snipers were deployed on rooftops, the group's director, Rami Abdul Rahman, told the AFP news agency.
State television later denied reports that helicopter gunships were being used in the assault. But pictures it broadcast of captured districts showed huge destruction, which activists say was cause by massive bombardments.
State TV also showed pictures of many bodies of what it called "terrorists" killed in the counter-attacks, as well as quantities of weapons and ammunition it said were seized.
Clearly, our correspondent says, in a straight fight, the lightly armed rebels are no match for the massive firepower and greater manpower of the state's forces.
Border crossing Reports from activists in Aleppo said there had been more clashes during the night and on Sunday morning between the rebel Free Syrian Army (FSA) and security forces.
They said a building in the Seif al-Dawla district collapsed under tank fire.
Residents also told the Reuters news agency that rebels were fighting government forces near the headquarters of an intelligence agency in the city centre.
An Aleppo-based activist, Mohammed Saeed, said rebels were still in control of the central Salah al-Din district and nearby Sakhour.
"Aleppo is witnessing serious street battles," he told the Associated Press.
Mr Saeed also said there had been clashes on the road leading to the city's international airport, as rebels tried to prevent military reinforcements being flown in.
State TV played down the scale of the violence, saying troops were merely hunting down "terrorists".
The commander of FSA forces in Aleppo province has vowed to "liberate" the whole city.
In a video posted online, Col Abdul Jabbar Mohammed Aqidi also called on government troops to defect to the opposition, and said the FSA would protect members of the president's minority Alawite sect, saying: "Our war is not with you but with the Assad family."
There were also reports of violence in the eastern city of Deir al-Zour on Sunday. Witnesses told Reuters that it was being attacked with artillery and rockets from helicopter gunships.
BBC sources in Syria also confirmed that rebels were now in control of the Bab al-Salam border crossing with Turkey. Turkey is not allowing non-Syrian nationals through so the border remains effectively closed.
President Assad was meanwhile shown on state TV receiving his new armed forces chief-of-staff, Gen Ali Abdullah Ayub, giving him his instructions, and wishing him success in his mission.