Syrian government sources said a total of 14 people died in the three bombings that hit two police stations and a central Damascus district mainly inhabited by members of President Bashar al-Assad's Alawite minority community.
The blasts came as the army pressed an offensive aimed at ousting rebels from footholds on the outskirts that they have used as launchpads for attacks.
A three-year-old child was among three people killed by the car bomb in the mainly Alawite neighbourhood of Mazzeh 86, the official SANA news agency reported.
Earlier, bombs hit a police station in Rokn Eddin in north Damascus and a security branch in Bab Musalla in the southwest, killing at least eight people, according to the Observatory.
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The interior ministry said 11 people died, six of them "terrorists", in the two attacks.
"Al-Nusra Front terrorists tried to enter the police station in Rokn Eddin and the criminal security branch in Damascus," it said.
"Three suicide attackers clashed with police in Rokn Eddin's police station as they prepared to detonate their explosives.
"Three other suicide bombers" tried to attack the criminal security branch in Bab Musalla, it added.
The ministry said investigations showed the attackers were members of Al-Nusra Front, a rebel group that has proclaimed allegiance to Al-Qaeda.
Prime Minister Wael al-Halqi condemned what he called "terrorist explosions targeting innocent citizens in Damascus" and said they showed the "desperation of the terrorists".
In northern Syria, 12 troops were killed in a rebel car bomb attack on the edges of Aleppo city, the Observatory said.
France's Hollande, on a visit to Qatar, urged the mainstream rebel Free Syrian Army to push groups like Al-Nusra out of the zones they control.