One of the suicide bombers on Sunday blew himself up at Tahrir Square near Bab Touma in central Damascus after being encircled by the authorities.
State TV said eight people were killed and at least a dozen others wounded in the attack.
The UK-based Syrian Observatory for Human Rights (SOHR), said 18 people were killed in the bombing, including at least seven members of pro-government security forces and two civilians. It had not identified the remaining victims.
Two other car bomb attackers were pursued and detonated by security forces on the road to Damascus airport southeast of the city.
State TV said the casualty toll had been minimised because the security forces had prevented "the terrorists from reaching their targets", saying they had aimed to target busy areas on the first day back to work after the Eid al-Fitr holiday.
Footage broadcast by state TV from Tahrir Square showed roads scattered with debris, several badly damaged cars, and another one that had been turned into a pile of twisted metal.
Mohamad Kheir Ismail, Damascus chief of police, in a phone interview with the Al Ikhbariya TV station, said the cars had been spotted on a highway and pursued.
He said the attackers were aiming for a big casualty toll but failed.
"This is the largest bombing to hit Damascus in about three months, and it came when life was starting to return to normal. People were returning to work and school after the Eid al-Fitr holiday," said Al Jazeera's Natasha Ghoneim, reporting from Gaziantep along Turkey's border with Syria.
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 city.
In mid-March, bomb attacks on a courthouse and restaurant in Damascus killed 32 people. They were claimed by the Islamic State of Iraq and the Levant (ISIL) group.
That came days after two explosions that left 74 dead in the capital's Old City and were claimed by the Hayat Tahrir al-Sham alliance, led by the former al-Qaeda affiliate Jabhat Fateh al-Sham.