It said security forces intercepted three suicide car bombers on Sunday morning, blowing up two at the entrance to the city.
The third was able to enter Damascus and detonated a bomb after being surrounded, causing several deaths and leaving a number of people wounded.
Syrian state news agency SANA quoted a senior police official saying that "terrorists simultaneously blew up three cars", two of them on the road to Damascus airport southeast of the capital and a third in the eastern Sahat Al-Tahrir district.
"The terrorist bombings killed and wounded several civilians and caused physical damage to the area," the official said.
Sahat Al-Tahrir resident Mohammad Tinawi told AFP he had heard "gunfire at around 6:00 am (0300 GMT), then an explosion which smashed the glass of houses in the neighbourhood."
He said he had seen Red Crescent volunteers treating two wounded soldiers, as well as two burned-out cars and damage to a security checkpoint.
Damascus has been spared the large-scale battles that have devastated other major Syrian cities during the country's six-year civil war.
But dozens of people have been killed in bombings, particularly on the outskirts of the capital.
In mid-March, bomb attacks on a courthouse and restaurant in Damascus killed 32 people. They were claimed by the Islamic State group.
That came days after two explosions that left 74 dead in the capital's Old City and were claimed by Tahrir al-Sham, led by the jihadist Fateh al-Sham Front.
Syria's conflict broke out with anti-government protests in 2011, but has since evolved into a multi-front war that has killed more than 320,000 people.
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