Syrian police had been chasing three suspected car bombers that were trying to enter the capital, state TV said.
Police stopped and detonated two of the vehicles, but the third driver entered Tahrir square in the east and blew himself up after being surrounded.
Syria is in the midst of a six-year-long civil war, with Damascus still mostly under government control.
At least 12 people were injured in Sunday's blast, reports said.
State TV said the attackers had planned to bomb crowded areas in the capital on the first working day after the Muslim holy month of Ramadan.
"The terrorist bombings killed and wounded several civilians and caused physical damage to the area," a police official told state news agency Sana.
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A local resident told AFP he heard "gunfire at around 06:00 (03:00 GMT), then an explosion which smashed the glass of houses in the neighbourhood".
An AFP correspondent at the scene saw extensive damage to nearby buildings and two bombed cars at one side of the square.
Syria's foreign ministry sent a letter to the UN saying up to 20 people were killed and dozens of women and children were among the wounded, Reuters reports.
No group has said it carried out the attack.
More than 300,000 people have lost their lives in the Syrian war, which began with anti-government protests in 2011.
The UN's refugee agency says that since the conflict began about 5.5 million people have left the country, and another 6.3 million have been left internally displaced.
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Damascus has remained mostly under the control of President Bashar al-Assad, and avoided much of the fighting.
However, the capital has experienced a number of suicide bomb attacks.
In March, two bomb attacks in the capital killed more than 40 people - the majority of them Iraqi pilgrims visiting the Bab al-Saghir cemetery, which houses Shia mausoleums. A jihadist group affiliated with al-Qaeda claimed that attack.
A few days later, an attack on the capital's main court complex killed at least 31 people. That attack was claimed by the Islamic State militant group (IS).
Such attacks may become more common as IS loses its territory and resorts to its tactic of striking soft targets in cities to sow instability, the BBC's Arab affairs editor Sebastian Usher reports.
The army is still fighting rebels in the eastern suburban areas of Jobar and Ain Tarma.
A suicide bomber has launched an attack in the Syrian capital, with reports saying at least 19 people were killed.