The explosion on Thursday was close to the Baath Party offices and the Russian Embassy, Syrian television said.
TV footage showed charred and bloodied bodies strewn across the street after the blast, which state media said was the result of a suicide bombing by "terrorists" battling President Bashar al-Assad.
Central Damascus has been relatively insulated from almost two years of unrest and civil war in which around 70 000 people have been killed across the country, but the bloodshed has shattered suburbs around the capital.
Rebels who control districts to the south and east of Damascus have attacked Assad's power base for nearly a month and struck with devastating bombs over the last year.
The al-Qaeda-linked rebel group Jabhat al-Nusra, which claimed responsibility for several of those bombs, says it carried out 17 attacks around Damascus in the first half of February, including at least seven bombings.
Casualties
Activists said most of the victims of Thursday's attack in the city's Mazraa district were civilians, including children, possibly from a school behind the Baath building.
Opposition activists reported further explosions elsewhere in the city after the explosion which struck shortly before 11am.
One resident in the heart of the capital heard three or four projectiles whistling through the sky, followed by explosions. At least one of them landed in a public garden in the Abu Rummaneh district, she said, but no one was hurt.
Embassy damaged The Britain-based Syrian Observatory for Human Rights, which monitors violence via a network of sources inside Syria, said the Mazraa car bomb detonated at a checkpoint close to the Baath Party building, located about 200m from the Russian embassy.
Russia's Itar-Tass news agency quoted a diplomat as saying the blast blew out windows at the embassy but no employees were wounded. "The building has really been damaged … The windows are shattered," the diplomat said. The vehicle was carrying between one and 1.5-tonnes of explosives, Damascus governor Bishr Sabban told Reuters.
Suicide bomber
A correspondent for Syrian television said he saw seven body bags with corpses at the scene. He counted 17 burnt-out cars and another 40 that were destroyed or badly damaged by the force of the blast, which ripped a crater 1.5-metres deep into the road.
Later, the Observatory reported, two car bombs exploded outside security centres in the northeastern district of Barzeh, but there were no details of casualties.
Syrian TV said security forces detained a would-be suicide bomber with five bombs in his car, one of them weighing 300kg.
In the southern city of Deraa, where the uprising against Assad erupted in March 2011, warplanes bombed the city's old district for the first time in nearly two years of conflict, killing 18 people, activists said.
A rebel officer in the Tawheed al-Janoub brigade which led a rebel offensive this week in Deraa said there were at least five air strikes on the city on Thursday.
"The (rebel) attacks on several major checkpoints in the Hay al-Saad neighbourhood and its declaration as a liberated area has prompted this response," said Abdullah Masalmah, an activist from the city, reached on Skype.
Fighting has intensified in southern Syria in recent weeks, leading to a sharp increase in refugee flows to neighbouring Jordan, according to officials. A Jordanian military source said 4 288 refugees arrived in the last 24 hours alone.
Nayef Hawatmeh, head of the Damascus-based Democratic Front for the Liberation of Palestine, was lightly wounded by an explosion in a mosque next to his office, a DFLP official said.
Talal Abu Tharifa, told Reuters in Gaza that glass fragments had caused a slight wound to Hawatmeh's hand. – Reuters