The news agency Sana said on Friday the group's leader, Abu Omar, and an accomplice were killed and another suspect was arrested.
It also said security forces busted the group following several months of surveillance after receiving information that a hitherto unknown group, Soldiers of Levant, was planning to carry out bomb attacks on various interests in an attempt to destabilise security in Damascus and its suburbs.
Sana said Syrian security troops had earlier discovered and detonated a roadside bomb on the Damascus-Zabadani road.
The news agency added that the group had also been planning what it called a terrorist attack inside the Justice Palace in Damascus.
According to the agency, a policeman was killed and another injured in the shootout.
"A gun battle took place between the security surveillance force and the terrorist group during which a member of the force was killed, another was injured, while two terrorists were killed and another was injured," Sana said.
Syrian authorities in the past
have faced armed rebellions
The police also found bombs, rifles, ammunition, mobile telephones and side arms when they stormed the apartment used by the group. It did not give further details about the clash, but said details will be published in the coming days.
The shootout - in the rural area of Daf al-Shawk near Damascus - is rare in Syria, which last witnessed a similar clash in April 2004.
A senior Syrian journalist has told Aljazeera that the group involved in the Damascus shootout was "an extremist criminal organisation that seeks to carry out sabotage activities against civilians and non-civilians in the country".
Talking to Aljazeera's correspondent in Damascus, the editor-in-chief of the Syrian al-Baath newspaper, Elias Murad, said on Saturday that the group wanted to incite violence and strife not only in Syria but also in other Arab societies.
Talking about Syria in particular, Murad said: "The more stable and nationalistic the Syrian stance becomes, the more it rejects demands dictated to serve Israel and other external forces.
He added: "Some people do not want Syria to live in stability. They want to influence the internal Syrian situation, so they could influence the decisions we take under all these pressures."