It was the first suicide bombing in Damascus since Bashar al-Assad was toppled by an Islamist-led rebel insurgency in December.
Syria's interior ministry said the suicide bomber was a member of Islamic State. He entered the church on Sunday, opened fire and then detonated his explosive vest, a ministry statement added.
A security source, speaking on condition of anonymity, said two men were involved in the attack, including the one who blew himself up.
Islamic State has been behind several attempted attacks on churches in Syria since Assad's fall, but this was the first to succeed, another security source told Reuters.
Syria's state news agency cited the health ministry as saying that 52 people were also injured in the blast.
A livestream from the site by Syria's civil defence, the White Helmets, showed scenes of destruction from inside the church, including a bloodied floor and shattered pews and masonry.
Syrian President Ahmed al-Sharaa, who led the offensive against Assad before taking over in January for a transitional phase, has repeatedly said he will protect minorities.
"We unequivocally condemn the abhorrent terrorist suicide bombing at the Mar Elias Greek Orthodox Church in Damascus, Syria," the Greek foreign ministry said in a statement.
"We demand that the Syrian transitional authorities take immediate action to hold those involved accountable and implement measures to guarantee the safety of Christian communities and all religious groups, allowing them to live without fear."
Islamic State had previously targeted religious minorities, including a major attack on Shiite pilgrims in Sayeda Zainab in 2016 — one of the most notorious bombings during Assad's rule.
The latest assault underscores the group's continued ability to exploit security gaps despite the collapse of its territorial control and years of counterterrorism efforts.
