The majority of dead were Iraqi pilgrims, according to the UK-based Syrian Observatory for Human Rights.
The Observatory put the death toll at 74, but the figure was not confirmed.
It took place near the Bab al-Saghir cemetery, which houses Shia mausoleums. Sunni militants often target Shias but attacks in the capital are uncommon.
Hayat Tahrir al-Sham (Liberation of the Levant Organisation) is a new group formed from Jabhat Fateh al-Sham (previously al-Nusra Front) and four smaller factions.
A nationwide truce in Syria, brokered by Russia, Turkey and Iran, took effect on 30 December after talks in Kazakhstan, but sporadic attacks continue.
There have been two rounds of talks, with another planned next week.
Damascus is mostly under the control of Syrian President Bashar al-Assad, but rebel groups are present in outlying districts of the city.
So-called Islamic State is another Sunni extremist group that controls significant areas of Syria. Its Iraqi stronghold of Mosul is currently under attack from Iraqi government forces and their allies.
More than 300,000 people have been killed and 11 million others displaced since the uprising against President Assad began in March 2011.