The SANA state news agency says "terrorists" were behind the attacks in the eastern district of Jaramana.
There has been fierce fighting in recent days around Damascus.
Reports say pro-government militiamen have set up armed groups in Jaramana, which is mostly populated by Christians and Druze, to defend the area against rebels.
Last month, a car bombing killed 11 people in the district.
Meanwhile, rebels in northern Syria claimed they brought down a government warplane near the rebel-held town of Daret Azzeh and captured its pilot.
Reports say the jet was shot down most likely by a missile.
Activists estimate that more than 40,000 people have died since the uprising against President Bashar al-Assad began in March 2011.