According to Syrian state-run TV, at least 50 people were killed and 170 were wounded when the explosions ripped through a military intelligence building.
Central Damascus has been under the control of forces loyal to President Bashar Assad since the rise of mass protests last year calling for political reform.
There was no claim of responsibility for the blasts. But an al Qaeda-inspired group has claimed responsibility for several past explosions, raising fears that terrorist groups are entering the fray and exploiting the chaos.
The U.N said weeks ago that more than 9,000 people had been killed. Hundreds more have died since the uprising.
The Associated Press contributed to this report.