State television said suspected al-Qaeda militants had targeted two security service bases in the Kafr Sousa area.
But the opposition said the attacks – which came a day after Arab League observers arrived – were staged by the government to justify its crackdown.
The Arab League team are tasked with monitoring whether the government complies with a peace agreement that orders all troops to withdraw from the streets, with the aim of ending the violence.
But human rights and opposition activists said the killings continued on Friday, with security forces shooting dead at least 12 civilians.
More than 5,000 people have been killed and thousands more detained since anti-government protests erupted in March, the UN says.
Within minutes, state television said two attacks had been carried out by suicide bombers driving vehicles packed with explosives against the General Security Directorate and another branch of the security services in the up market Kfar Sousa district, south-west of Damascus centre.
“Preliminary investigations showed al-Qaeda was responsible,” it added.
Video footage was broadcast of heavily damaged buildings, with rescue workers combing through burnt buildings and blood-stained debris, and ambulances taking the injured away.
The state-owned news channel, al-Ikhbariya al-Suriya, said the first car bomb exploded outside the offices of an unspecified security agency.
When guards at a nearby compound housing the General Security Directorate went to inspect the aftermath of the first blast, the driver of another vehicle rammed the main gates and detonated the bomb it was carrying, the channel said.
“The explosions shook the house, it was frightful,” Nidal Hamidi, a Syrian journalist who lives in Kfar Sousa, told the Associated Press news agency.
“Gunfire was heard immediately following the explosion and windows up to 200m (670ft) away were shattered,” Nidal Hamidi said.
Deputy Foreign Minister Faisal Mekdad told reporters: “On the first day after the arrival of the Arab observers, this is the gift we get from the terrorists and al-Qaeda.
“But we are going to do all we can to facilitate the Arab League mission.”
Faisal Mekdad was accompanied by the Arab League’s Assistant Secretary General, Samir Seif al-Yazal, who said the nine-strong advance team of monitors would not be deterred.
“We are here to see the facts on the ground,” he added.
“What we are seeing today is regretful; the important thing is for things to calm down.”
The US state department condemned the attacks in Damascus but said they must not deter the Arab League observers from doing their work.
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