The agency, quoting police, said the twin attack took place at a security checkpoint near the shrine of Sayyida Zeinab, a granddaughter of the Prophet Mohammed.
"A civilian car had stopped at the checkpoint... and a terrorist got out wearing an explosive vest which he detonated," SANA said, adding that seconds later a man who had remained in the car blew it up.
The Syrian Observatory for Human Rights said the bombing targeted a checkpoint manned by a pro-regime militia, just 10 kilometres (six miles) east of Damascus and on the road to the international airport.
Signup to our newsletter and follow us on Facebook and Twitter!
On February 1, a blast ripped through a bus carrying Lebanese Shiite pilgrims in the city, killing at least nine people in an attack claimed by Al-Qaeda's Syrian branch.
At the time, the bus had been heading to Sayyida Zeinab which is controlled by the Syrian army and allied militias.
The Britain-based Observatory also reported that a rocket hit the Adra prison northeast of Damascus on Monday, killing four people and wounding 15.
It did not say if the casualties were prisoners, guards or visitors.