"At least 10 soldiers were killed and at least 10 others were injured," Rami Abdel Rahman, director of the Britain-based Syrian Observatory for Human Rights, told AFP.
"There was a first major blast from a car bomb, and then smaller explosions that appeared to be from rockets fired at the area subsequently," he added.
The attack was carried out on an army checkpoint near the Mazzeh military airport, in the western suburbs of Damascus.
The explosion was heard across parts of the Syrian capital, and flames were visible from a distance, said the Observatory, which relies on a network of activists, medics and lawyers on the ground for its information.
The military airport is guarded by the army's elite Fourth Division, which is responsible for security in and around Damascus.
The division is led by Maher al-Assad, the brother of President Bashar al-Assad whose regime has been battling to crush an uprising for more than two years.
A series of attacks have hit Damascus in recent months, most of them aimed at security checkpoints, soldiers or government buildings.
On April 29, a car bomb exploded in Mazzeh targeting the car of Prime Minister Wael al-Halqi. He escaped unscathed but his bodyguards were killed.
More recently, twin suicide bombers targeted a police station in the city centre on Tuesday last week, killing at least 14 people and wounding more than 30 others.
The conflict pitting Assad's regime, which is dominated by his Alawite sect of Shiite Islam, against Sunni Muslim-led rebel fighters is estimated to have killed at least 93,000 people since it erupted in March 2011.