"On Saturday... A twin attack was carried out by two heroes of Islam... In the centre of the capital Damascus, killing and wounding dozens," the statement said.
Without elaborating, it called the bombings "a message to Iran and its militias".
Iran and the Tehran-backed Hezbollah movement of Lebanon are close allies of the Syrian government.
The Syrian Observatory for Human Rights said 43 Iraqi pilgrims were among those killed when a roadside bomb detonated as a bus carrying pilgrims made its way through the Ban al-Saghir area of Damascus's famed Old City and a suicide bomber blew himself up.
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State television gave a toll of 40 killed and 120 wounded by "two bombs detonated by terrorists", while Iraq's foreign ministry said around 40 of its nationals had died.
In January, Fateh al-Sham Front also claimed responsibility for twin suicide bombings in the high-security Kafr Sousa district of Damascus that killed 10 people, eight of them soldiers.
Damascus, stronghold of President Bashar al-Assad's government, has been largely spared the violence that has rocked other major cities in Syria's six-year war.
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