A local resident told AFP he heard “gunfire at around 06:00 (03:00 GMT), then an explosion which smashed the glass of houses in the neighbourhood”.
Image captionPolice said the attackers had intended to target busy areas in the capital
No group has said it carried out the attack.
More than 300,000 people have lost their lives in the Syrian war, which began with anti-government protests in 2011.
The UN’s refugee agency says that since the conflict began about 5.5 million people have left the country, and another 6.3 million have been left internally displaced.
Damascus has remained mostly under the control of President Bashar al-Assad, and avoided much of the fighting.
However, the capital has experienced a number of suicide bomb attacks.
In March, two bomb attacks in the capital killed more than 40 people – the majority of them Iraqi pilgrims visiting the Bab al-Saghir cemetery, which houses Shia mausoleums. A jihadist group affiliated with al-Qaeda claimed that attack.
Such attacks may become more common as IS loses its territory and resorts to its tactic of striking soft targets in cities to sow instability, the BBC’s Arab affairs editor Sebastian Usher reports.
The army is still fighting rebels in the eastern suburban areas of Jobar and Ain Tarma.