The state-TV Ikhbariya TV station showed footage of broken shop facades and mangled cars in the central Marjeh Square as ambulance workers were seen carrying the wounded on stretchers. The bustling square was scene to previous attacks earlier this year.
Syrian state TV said two civilians were killed because of the explosions caused by two "terrorist" suicide bombers. It said another 26 were wounded.
The Britain-based Syrian Observatory for Human Rights, which has a network of activists on the ground in Syria, confirmed the death toll but said the explosions were caused by bombs laden in the area. It was not immediately possible to reconcile the two accounts.
Suicide attacks and car bombs have become common in Damascus. Tuesday's twin explosions in the capital are the first since government troops captured Qusair, a strategic town in the central province of Homs, the linchpin linking Damascus with the regime strongholds on the Mediterranean coast.
There was no immediate claim of responsibility, but car bombs and suicide attacks targeting Damascus and other cities that remain under government control have been claimed in the past by the al-Qaida affiliated Jabhat al-Nusra — one of scores of rebel factions fighting the forces of President Bashar Assad.
On Saturday, a suicide bomber detonated his explosives-laden car in the central city of Homs, tearing through an area largely populated by the regime's Alawite sect and killing seven people.
Syria's conflict started with largely peaceful protests against Assad's regime in March 2011 but eventually turned into a civil war that has killed more than 80,000 people, according to the United Nations.