The al-Nusra Front, a jihadist militant group, said explosives-filled vehicles had been blown up outside the Air Force Intelligence complex in Harasta.
Residents said there were several huge blasts, followed by fierce clashes.
One activist group said dozens of people were killed and expressed fears for the fate of the agency's prisoners.
Human rights activists say hundreds of opponents of President Bashar al-Assad have been detained and tortured by Air Force Intelligence (AFI) across the country since the uprising began in March 2011.
AFI is seen by some as the elite agency of Syria's intelligence empire.
Though smaller than Military Intelligence, in the past it has played a leading role in operations against Islamist opposition groups, as well as covert actions abroad, and has a reputation for brutality.
'Citadel of repression'
The group, which has claimed to be behind a series of deadly bombings since January, said a vehicle packed with nine tonnes of explosives had been driven up to the AFI compound in Harasta and blown up.
Twenty-five minutes later, another al-Nusra Front fighter blew up a one-tonne bomb hidden inside an ambulance, targeting survivors, it added. The fighters then fired mortars at the complex, the statement said.
The Syrian Observatory for Human Rights (SOHR), a UK-based activist group, reported that several large explosions had shaken homes overnight in Harasta, as well as in the nearby suburbs of Jobar and Qabun.
The blasts were followed by violent clashes in several parts of Harasta, with government forces firing heavy weapons and mortars, it added.
SOHR head Rami Abdul Rahman told the AFP news agency that dozens of people had been killed in the suicide attacks.
"The fate of hundreds of prisoners being held in the basements of the [security complex] is still unknown," he added. "The regime has not said a word about what happened last night."
The SOHR is one of the most prominent organisations documenting and reporting incidents and casualties in the Syrian conflict. The group says its reports are impartial, though its information cannot be verified.
A Syrian official told the Associated Press that a suicide bomber had detonated a car bomb near the AFI complex, while pro-government al-Ikhbariya TV reported that the blast was followed by clashes.
Elsewhere on Tuesday, government forces continued to attack rebel strongholds in the central city of Homs.
State TV reported that soldiers had secured large parts of the Khalidiya district and were now "pursuing the remnants of the terrorists".
News of the violence came as UN Secretary General Ban Ki-moon urged the Syrian government to immediately declare a unilateral ceasefire, and the opposition to abide by it.
"It is unbearable for the Syrian people to continue like this," he told a joint news conference in Paris with French President Francois Hollande.
Mr Ban also called on countries to stop supplying weapons to both sides.
On Monday, the BBC found evidence that weapons intended for the Saudi military have been found at a base used by Syrian rebels.
The BBC's Ian Pannell saw crates from a Ukrainian arms manufacturer - addressed to Saudi Arabia - at a rebel camp in the city of Aleppo.
Meanwhile, Nato Secretary-General Anders Fogh Rasmussen has said the alliance has "all necessary plans in place to protect and defend Turkey if necessary", amid escalating tensions on its border with Syria.
Turkish and Syrian government forces have been exchanging artillery fire since Syrian shells killed five Turkish civilians last week.