Syria blamed “terrorist groups linked to al-Qaeda” for the blast, which hit a central district near the headquarters of Syria’s ruling Baath Party.
TV pictures showed images of bodies, wrecked cars and shattered windows.
The violence comes as Russia and the Arab League say they want to broker direct government-opposition talks.
No group has yet admitted the attack.
The Syrian foreign ministry blamed the bombing on “armed terrorist groups linked to al-Qaeda that receive financial and logistic help from abroad”.
The militant Islamist al-Nusra Front previously said it had carried out many of the bombings that have rocked Syria since the uprising began in March 2011.
Meanwhile Russian Foreign Minister Sergei Lavrov described the war as “a road to nowhere”.
The opposition Syrian National Coalition is holding a two-day meeting in Egypt to discuss a framework for a possible solution.
Also on Thursday, the UN and Arab League envoy for Syria, Lakhdar Brahimi, agreed to remain in office through the rest of 2013, UN spokesman Martin Nesirky told the Associated Press. Mr Brahimi’s contract was due to expire on Friday.
The UN has estimated up to 70,000 people have died in the conflict in the past two years.
Police and witnesses said the blast was a car bomb. It went off in the central Mazraa neighbourhood, close to the Baath offices and Russian embassy.
State and pro-regime TV showed pictures of dead bodies and destroyed cars. State media said at least 53 were killed and another 200 injured in the blast. The Syrian Observatory for Human Rights, a UK-based activist group, said at least 42 had died, most of them civilians.
Surrounding roads are reported to have been closed off to traffic and firefighters and medical staff were soon at the scene.
Witnesses told AP news agency the car had exploded at a security checkpoint between the Russian embassy and the Baath Party central headquarters.
“It was huge. Everything in the shop turned upside down,” one local resident said. He said three of his employees were injured by flying glass that killed a young girl who was walking by when the blast hit.
“I pulled her inside the shop but she was almost gone. We couldn’t save her. She was hit in the stomach and head.”
State media said the explosion had struck near a school and clinic and that schoolchildren were among the casualties.
It seems to have been targeted at the Baath party offices, but also affected residential areas, says the BBC’s Lina Sinjab in Damascus.
Heavy fighting between government and rebel forces continued around the city, with the government carrying out air strikes in the suburbs.
Shortly after the car bomb, two mortars were fired at a military headquarters in Damascus, reports say.
And there were two other explosions in the city, also at security checkpoints, according to the SOHR.
The UK-based activist group is one of the most prominent organisations documenting and reporting incidents and casualties in the Syrian conflict. The SOHR says its reports are impartial, though its information cannot be independently verified.
Mr Lavrov said the Kremlin and the Arab League wanted to establish direct contact between the Syrian government and the opposition.
Speaking in Moscow, where he hosted league officials and several Arab foreign ministers, the Russian foreign minister said that sitting down at a negotiating table was the only way to end the conflict without irreparable damage to Syria.
“Neither side can allow itself to rely on a military solution to the conflict, because it is a road to nowhere, a road to mutual destruction of the people,” he said.
Mr Lavrov and Arab League General Secretary Nabil Elaraby said their priority was to create a transitional government to navigate a way out of the violence.
No conditions for the negotiations have been set, they said.
The proposal initially received a cool reception from the Syrian National Coalition (SNC), with senior member Abdelbaset Sieda insisting Mr Assad and his allies “must go first”.
“After that we can discuss with others in the regime who didn’t share in the killing of our people,” he said.
But the news agency Reuters says it has seen a draft SNC communique being discussed in Cairo which demonstrates an apparent softening in the group’s stance.
The document reasserts the group’s position that Mr Assad’s apparatus cannot be part of any political solution in Syria, but omits previous demands that Mr Assad’s regime must go even before any talks, Reuters says.
But that may still prove unacceptable in Damascus, says the BBC’s James Reynolds in Istanbul.