Three attackers had attempted to stage bombings using cars laden with explosives, Syrian authorities said.
Security forces reportedly chased and intercepted two further suspected vehicle bombers on the road to Damascus airport as they attempted to enter the city, before releasing the explosives.
Police managed to stop and detonate two of the vehicles, but the third driver entered Tahrir square in the east and blew himself up after being surrounded.
Damascus was hit by two separate, multiple suicide bomb attacks in March, one of them claimed by Islamic State and the other by the Islamist insurgent alliance Tahrir al-Sham.
Damascus chief of police Mohamad Kheir Ismail, in a phone interview with the al-Ikhbariya TV station, said the cars had been spotted on a highway and pursued.
No group has claimed responsibility for the Sunday attacks.
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The source, speaking on condition of anonymity, said the terrorists meant to detonate the cars inside crowded areas in the capital, as people were back to work after a week-long holiday of Eid al-Fitr.
Pro-government forces have engaged in heavy fighting in Damascus' suburbs during the war, but have largely kept the rebels out of the city center.
Damascus has been spared the large-scale battles that have devastated other major Syrian cities during the country's six-year civil war.
The opposition monitoring group said the number of dead is likely to rise due to the number of critically injured.
Several individuals are believed to have been killed or injured by the blast.
Damascus has remained under the control of President Bashar al-Assad and avoided much of the fighting, but it has experienced a number of suicide bomb attacks.