State media said troops had foiled an attack by a "terrorist group" in the Rukn al-Din district, where several major security facilities are located.
The UK-based Syrian Observatory for Human Rights said the jihadist group al-Nusra Front had targeted an army general responsible for logistics.
Maj Gen Mohammed Eid was injured, and one of his guards was killed, it added.
However, security sources denied that the general had been targeted.
Monday's attack took place near a building used by the Syrian army's munitions and supply division.
Witnesses told the AFP news agency that the blast emanated from a small motorcycle, and that it was followed by 15 minutes of gunfire.
"A terrorist group rode into the Rukn al-Din neighbourhood on motorcycles and began clashing with security forces," a security official told AFP.
"When they realised they could not escape, one of them detonated a suicide belt he was wearing," he said, adding that security forces killed the remaining rebels.
A local hospital said it was treating six people wounded by the explosion, but did not identify them.
Rami Abdul Rahman, the head of the Syrian Observatory for Human Rights, said a member of al-Nusra Front had blown himself up next to a car carrying Gen Eid as it passed through Rukn al-Din.
He said Gen Eid and two of his guards were among those injured, but that was denied by the security official.
Al-Nusra Front, al-Qaeda's affiliate in Syria, later claimed responsibility for the attack on Twitter, saying three of its fighters had been "able to infiltrate the logistics building".
Many senior officials live in Rukn al-Din and a number of branches of Syria's intelligence apparatus are based there, according to Mr Abdul Rahman.
A resident of Rukn al-Din told Reuters news agency that the army had sealed off main roads in the area after the incident and detained scores of people.
The heavily-defended centre of Damascus has seen several major bombings since the start of the uprising against President Bashar al-Assad in 2011, but suicide bombings are rare.
In 2012, the defence minister, the president's brother-in-law and the head of his crisis team were killed by a bomb blast at the National Security Bureau's headquarters in the capital.