The neighborhood, home to mostly Christian and Druze residents, is considered supportive of President Bashar Assad's government and has been targeted previously, The Washington Post reported.
The explosions occurred a day after rebel fighters seized two military bases in Syria, the latest in their capture of several key government military installations.
The Syrian Network for Human Rights said it had documented the deaths of 104 people across Syria Wednesday, including 57 by the car bomb in the Jarbana neighborhood. A second car bomb and other fighting killed 76 people, including four children, in Damascus and its suburbs. Elsewhere, 10 died in Daraa, six each in Homs and Idlib, five in Aleppo and one in Dier al-Zoor.
Video of bloody streets and demolished cars at the explosion site in Jarbana was aired on state-run television and the official Syrian Arab News Agency blamed "terrorists" -- the characterization used by state media for opposition fighters -- for the attack.
The BBC reported no group has claimed responsibility for the blasts.
SANA reported two devices were detonated elsewhere in Damascus but only caused "minor material damage."
Elsewhere Wednesday, Syrian fighter jets attacked rebel positions in the western Damascus suburb of Darayya, the Syrian Observatory for Human Rights, based in Britain, said.
The army also reportedly shelled Zabadani, northwest of the capital, the BBC said. The Syrian Revolution General Commission, an opposition network, said more than 50 shells rained on the town in a 30-minute period, injuring several people.
Both of the bases rebels captured Tuesday were used by the Syrian air force, opposition groups said. One base was south of Damascus and the second base was southeast of Aleppo.
In Idlib province Tuesday, Syrian warplanes attacked an olive press, killing at least 20 people and wounded dozens of others, the Local Coordination Committees, another activist network, said.
One observer told the Post it was uncertain how rebel fighters will be able to hold the military base they captured because they've been losing territory to the Syrian military as well.
"They've lost many areas in eastern Syria and near Aleppo recently," Syrian Observatory Director Rami Abdul Rahman said. "It's difficult to see a serious change."