The men approached the embassy compound on Tuesday morning and then attempted to blow up a car outside, according to Syrian television.
A Syrian official said all US diplomats were safe while Syria’s Sana news agency reported that a member of the Syrian anti-terror squad was killed while defending the building.
A Chinese diplomat was also slightly wounded by a stray bullet, according to Chinese state media.
Syrian state television said the attackers had tried to detonate a car bomb in front of the embassy, but had failed.
The men were reportedly chanting religious slogans as they attacked the building.
Television footage of the scene showed a van packed with gas canisters and detonators taped to them, as well as bloodstains on the pavement and a bullet-riddled car.
Syrian security forces sealed off al-Rawda area after heavy gunfire started.
The district also houses security installations and the homes of senior government officials.
There was no word on the identity of the attackers, but Syrian forces clashed with Islamist fighters several times last year, usually in raids to arrest them.
Car bombing foiled
The interior minister described the attack as “a terrorist operation”, and told Syrian television that “investigations are under way to find out more details”.
The US secretary of state, Condoleezza Rice, said: “It’s too early to tell who might have been responsible for the attack… We will have to do the forensics.”
Rice thanked Syrian security forces and expressed condolences over the death of the security officer.
“I do think that the Syrians reacted to this attack in a way that helped to secure our people, and we very much appreciate that,” she said.
This is the first time an embassy
Meanwhile, speaking to Aljazeera on Tuesday from the Syrian capital, Mahdi Dakhlallah, the former Syrian information minister, said: “Damascus has witnessed similar incidents before, but this is the first time an embassy has been targeted.
“The authorities have always tried to maintain a balance between the demands of security and citizens’ freedom of movement.
“Some say that the embassy area is a security zone, but this is not true. People can freely move around any embassy in Damascus despite the heavy security arrangements in the area.”
Dakhlallah continued: “What happened on Tuesday could happen in any other place in the world.
“The whole region is witnessing terrorist attacks. Terrorism level in the region has increased by around 40 per cent over what it was before the war.
“We hope the US will understand that its unjust policies in the region are leading to unrest and giving terrorist groups justification for carrying out attacks against Arab countries.
Nevertheless, Dakhlallah said the security situation in Syria is by and large stable, with a few exceptional cases.