"Families are trying to flee their homes, but it is difficult to get out of the neighbourhood. It is surrounded, and violence on the edges is intense," he said.
Abu Omar also said the army raided the nearby Rukn al-Din neighbourhood, while "helicopters used machineguns to fire into the district's streets."
The Syrian Observatory for Human Rights said "the feared Fourth Brigade" commanded by President Assad's powerful younger brother Maher was carrying out the Barzeh attack.
"Troops have stormed the northwestern Barzeh district of Damascus with tanks and armoured personnel carriers," the group's director Rami Abdel Rahman said, adding that snipers were deployed on rooftops.
The rebel Free Syrian Army's (FSA) military council head General Mustafa al-Sheikh told AFP "a real war of attrition" was underway in Damascus.
"The regime is collapsing, the speed at which it is falling has increased. That means it will use greater violence in order to try and save itself," said Sheikh.
The Observatory also warned that a siege was underway on the outskirts of the upscale Mazzeh neighbourhood, saying "dozens of tanks" were preventing medical teams from reaching scores of injured.
Nationwide, 94 people were killed in violence on Sunday, 70 of them civilians, the watchdog said. It added that over the past seven days as many as 1,261 people have been killed, three quarters of them civilians.
The official SANA news agency announced that government forces had "cleansed" the capital's Qaboon neighbourhood of "terrorists," the regime's term for rebel fighters.
And state television aired footage reportedly from Qaboon showing dead bodies and weapons, communications equipment and money it said was captured from rebels.
It said some of the rebels killed held identity cards of Jordan and Egypt, accusing foreign countries of training and sending in insurgents.
But it denied helicopter gunships were being used inside the capital.
Residents in the city reported bread shortages, and the UN agency for Palestinian refugees UNRWA expressed concern about the fate of residents of the Yarmuk camp in the city.
As battles continued in the capital, a rebel FSA commander declared that the battle to "liberate" the northern city of Aleppo had begun.
In a YouTube video, Colonel Abdel Jabbar Mohammad Oqaidi announced "the start of the operation to liberate Aleppo from the hands of Assad's gangs," while pledging that rebels would protect civilians, including the city's minorities.
Fierce clashes engulfed its Salaheddin and Sakhur districts, with an anti-regime activist saying the army began an assault at dawn. SANA said soldiers were "chasing down terrorists," adding that "many of them were killed."
In a new toll, the Observatory said that more than 19,000 people have been killed since the uprising against Assad's regime began in March 2011.
On Syria's borders, rebels battled troops for control of crossing posts with Turkey, Iraq and Jordan, as Turkey moved batteries of ground-to-air missiles to its frontier with the Arab state.
By Sunday afternoon, fighters were in control of three crossings with Turkey -- Bab al-Hawa, Al-Salama and Jarabulus. They also controlled Albu Kamal post with Iraq but lost hold of Rabiyah on the same border a day after capturing it.
At the Al-Salama crossing, on the Turkish border north of Aleppo, an AFP photographer saw about 17 rebel fighters in control. They had set fire to the portraits of Assad normally hung on the walls.
The rebel commander at the post, Ammar Dehdeh, said the fighting had lasted some two hours. He said his fighters had killed nine government troops and captured 20.
Fighting has intensified since a Wednesday bombing that killed national security chief General Hisham Ikhtiyar, Defence Minister General Daoud Rajha, Assad's brother-in-law Assef Shawkat and General Hassan Turkmani, head of the regime's crisis cell on the uprising.
With the violence escalating, more than 2,000 Syrians fled to Jordan early on Sunday, a prominent local charity in the neighbouring Arab state said, expecting a largescale influx.
Thousands of refugees have also crossed into neighbouring Lebanon and Turkey, where stone-throwing Syrians on Sunday clashed with police at two camps over a lack of food and water.