State media said security forces intercepted the two other car bombs, suggesting they were controlled explosions.
Footage on state TV from the scene of one explosion, along the road to the airport, showed at least two scorched vehicles. The closed-off street was littered with debris, while security men roamed the area.
Footage from Tahreer Square in central Damascus showed the facade of one building badly damaged, and mangled vehicles parked in the small roundabout.
The state news agency SANA said the security forces detonated two car bombs at the airport road intersection.
More From This Section
The Britain-based Syrian Observatory for Human Rights, an opposition-run monitoring group, also reported the three explosions. It put the death toll at 12, including the suicide bomber.
Such attacks have been relatively rare in Damascus, the seat of power for President Bashar Assad.
Pro-government forces have been fighting to drive rebels from Ain Terma, one of their last strongholds on the eastern outskirts of Damascus.