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8 year oldIslamic State group said three militants - two carrying explosives belts and one with a car bomb - carried out the attacks in Sayyida Zeinab, 10km (six miles) south of the centre.
At least 13 other people were wounded.
Sayyida Zeinab, a predominantly Shia area, has been a frequent target of attacks f-rom jihadi groups.
The Sunni IS group says it was behind at least two other bombings in the district earlier this year that killed more than 150 people.
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Syrian state media said Sayyida Zeinab was hit by two blasts on Saturday - a suicide attack at the entrance to the district and then a car bomb attack in al-Teen street, not far f-rom the shrine.
The IS-affiliated Amag news agency later said three IS militants carried out the attacks.
The golden-domed Sayyida Zeinab shrine contains the grave of one of the Prophet Muhammad's grand-daughters, and continues to draw many Shia pilgrims despite the civil war.
Shia fighters f-rom around the region have joined the conflict in Syria on the grounds that they wish to protect the shrine f-rom the civil war, observers say.
The Lebanese Shia movement Hezbollah has cited it as a key reason that it chose to fight on the side of Syrian President Bashar al-Assad.
But this has made this suburb a prime target for the government's enemies, including IS, the BBC's Alan Johnston says.
More than 250,000 people have died and 11 million have fled their homes in more than five years of civil war in Syria.