Several Shiite pilgrims were among the dead.
"There are also dozens of people wounded, some of them in a serious condition," Observatory chief Rami Abdel Rahman told AFP.
The director general of the capital's Al-Mujtahed hospital told AFP earlier there were at least 28 people dead and 45 wounded.
The SANA state news agency reported that "two bombs planted by terrorists exploded near the Bab al-Saghir cemetery in Bab Musalla, causing dead and wounded."
Shiite shrines are a frequent target of attack for Sunni extremists of Al-Qaeda and the Islamic State group (IS), not only in Syria but also in neighbouring Iraq.
The Sayeda Zeinab mausoleum to the south of Damascus, Syria's most visited Shiite pilgrimage site, has been hit by several deadly bombings during the six-year-old civil war.
Twin suicide bombings in the high-security Kafr Sousa district of the capital in January killed 10 people, eight of them soldiers.
That attack was claimed by former Al-Qaeda affiliate Fateh al-Sham Front which said that it had targeted Russian military advisers working with the Syrian army.
It was widely seen as an attempt to disrupt UN-brokered peace talks that took place the following month which to the anger of Fateh al-Sham were supported by its former Islamist rebel ally Ahrar al-Sham.
Fateh al-Sham has been repeatedly bombed in its northwestern stronghold this year, not only by the Syrian army and its Russian ally but also by a US-led coalition battling IS in both Syria and Iraq.
The rift over the UN-brokered talks between the rebels and the government has also seen deadly clashes between the jihadists and their former Islamist rebel allies.
The two groups had together seized virtually all of the northwestern province of Idlib but are now vying for territorial control.
Bomb attacks are rare in Damascus, a stronghold of the regime of President Bashar al-Assad.
The Syrian capital is sometimes the target of shelling by rebel groups who hold areas on the outskirts.
On December 16, a seven-year-old girl wearing an explosive belt blew herself up outside a police station in Midan district, wounding three police officers.
Two blasts near state security agencies in Kafr Sousa in December 2011 killed more than 40 people and wounded more than 150, the Syrian government said at the time.
(This story has not been edited by NDTV staff and is auto-generated from a syndicated feed.)