The blasts, one of which was caused by a car bomb, happened near the Shia Muslim shrine of Sayyida Zeinab on Friday, the interior ministry said.
State television footage showed burning buildings and car wreckage in the neighbourhood where the shrine is located.
The Islamic State of Iraq and the Levant (ISIL) claimed responsibility for the attack.
The shrine has been targeted before, including in February 2015 when two suicide attacks killed four people and wounded 13 at a checkpoint nearby.
In the same month, a blast ripped through a bus carrying Lebanese Shia Muslim pilgrims headed to Sayyida Zeinab, killing at least nine people in an attack claimed by the al-Nusra Front armed group.