The assault was the second time in a month that suicide attackers have targeted the capital, which has been insulated from the worst of the violence in the war-torn country.
State news agency SANA quoted Interior Minister Mohammad Shaar saying both of those killed were police officers.
Two of the suicide bombers detonated their explosives in front of the police HQ on Khaled Bin al-Walid street in central Damascus, his ministry said in a statement carried by state media.
"The terrorist suicide attackers tried to storm the police command headquarters... The guards opened fire on them, forcing them to blow themselves up before they entered the building and achieved their goals," it added.
Police surrounded a third attacker behind the building who also blew himself up, the statement said.
The interior ministry said two people had been killed and six wounded in the attack, among them two children.
The Britain-based Syrian Observatory for Human Rights reported five dead, not including the three attackers.
IS in its statement on Telegram said two attackers "entered the headquarters building and fought with those inside... then detonated their explosive vests".
Damascus police chief Mohammed Khairu Ismail told reporters at the scene in the central neighbourhood of Qanawat that one of the dead was a policeman who tried to stop the bombers.
Officer 'grabbed bomber'
"One of our forces grabbed one of the suicide bombers and prevented him from entering the building, so he blew himself up, killing the sergeant," Ismail said.
It was the second time this month that suicide attackers have targeted police in the capital. At least 17 people were killed in an October 2 attack on a police station in the southern district of Midan.
That attack was also claimed by IS, which said three fighters armed with guns, grenades and explosives had targeted the station.
More than 330,000 people have been killed in Syria since the country's conflict began in March 2011 with anti-government protests.
Damascus has been shaken by several bomb attacks, despite being largely spared from the worst of the violence in the six-year war.
In this month's incident at Midan, one attacker was able to reach the first floor of the police station before blowing himself up.
The Midan police station itself had previously been targeted in December 2016, when a seven-year-old girl entered the building wearing an explosive belt that was remotely detonated.
Rebel groups have been gradually expelled from territory they once held in the capital, though they maintain a presence in a handful of positions, including the Jobar neighbourhood.
They also hold territory in the Eastern Ghouta region outside the capital, and have regularly launched rockets into the city.
A deal reached between rebel backer Turkey and regime allies Russia and Iran earlier this year has created so-called "de-escalation zones" in several parts of the country, including Eastern Ghouta, bringing a measure of calm.
But the fight against IS continues, with government troops and the US-backed Syrian Democratic Forces, a coalition of Kurdish and Arab fighters, leading separate offensives against the jihadists in northern and eastern Syria.
