State media blamed the blast near the headquarters of the ruling Baath party on "terrorists".
TV pictures showed images of dead people. Overseas activists said at least 31 people had been killed.
The violence comes as Russia and the Arab League say they want to broker direct government-opposition talks.
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.
Some 70,000 people have died since the uprising against President Bashar al-Assad began in March 2011, the UN says.'Upside down'
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.
Surrounding roads are reported to have been closed off to traffic and firefighters and medical staff were at the scene.
Images from state media showed wounded Syrians reeling from the blastState and pro-regime TV showed pictures of dead bodies and destroyed cars.
The UK-based Syrian Observatory for Human Rights said at least 31 people had been killed in the explosion.
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.
There have been two other explosions in the city, also at security checkpoints.
And heavy fighting between government and rebel forces is continuing around the city, with the government carrying out air strikes in the suburbs.
The BBC's Jim Muir in Cairo says although the Syrian government and the opposition are talking about dialogue, that still seems a distant prospect.
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.Opposition 'softens'
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.
