At least 30 other people were injured in the blasts, in Marjeh Square.
The explosions happened at a police building in the busy commercial district.
The attack comes as regime forces prepare an assault to recapture the northern city of Aleppo after having retaken Qusair from the rebels.
Images on Syria's al-Ikhbariya TV showed a scene of widespread damage, with shop fronts blown out and debris littering the street.
Blood stains marked the pavement, while people milled around among broken glass and wreckage.
One bomber reached the first floor of the police station and detonated his explosives when he came under fire, police told the BBC's Lyse Doucet in Damascus.
Several walls collapsed from the force of the blast. Others are splattered with blood and flesh, and punctured by holes from ball bearings packed inside the bomb.
The entire floor is littered with broken furniture and chunks of plaster, police sources say.
Police say a second suicide bomber exploded his device shortly afterwards, on the street below, as police were leaving the station.
It is unclear whether police are among the dead and injured.
Such attacks are not uncommon in Damascus - the same square was struck just six weeks ago.
But residents of the city centre say the situation there is much quieter now than it was a few months ago, when rebel forces were pressing in around the city and a final assault seemed imminent.
Since then, a major counteroffensive by regime forces has pushed the rebels back from the nearby suburbs, especially on the eastern and southern flanks of the capital.
Syria's conflict started more than two years ago, with largely peaceful protests against the regime of President Bashar al-Assad.
But it has now turned into a civil war that has killed more than 80,000 people, according to the United Nations.
Sights on Aleppo Syrian government forces are reported to be preparing for a major offensive on rebel-held parts of Syria's largest city, Aleppo.
Opposition activists in Aleppo told the BBC that military reinforcements - including fighters from Hezbollah, the Lebanese Shia Islamist group - had already been sent to parts of the city.
Aleppo has been carved up into areas controlled by rebels and the government
On Sunday, they retook the last remaining rebel-held villages
in the strategically important area, which lies between the Lebanese
border and the central city of Homs.
Much of northern Syria has been controlled by rebel groups since last year, and the front lines inside Aleppo have been largely static for months.
But the fall of Qusair last Wednesday has been seized on by the military leadership in Damascus as a decisive victory.
In view of the Syrian regime's advance, Washington could decide this week to start arming the rebels, US officials say.
A State Department spokeswoman said on Monday that the US would "continue to look for ways to help the opposition and increase aid... The president has talked about how boots on the ground is not an option - so all options short of that".
Boy's 'execution'
In a separate development on Sunday, the Syrian Observatory for Human Rights (SOHR) reported that Islamist rebels in Aleppo had executed a 15-year-old boy in front of his parents as punishment for what they regarded as a blasphemous comment.
The UK-based activist group said Mohammed Qataa was shot in the face and neck a day after being seized by the rebels, who allegedly overheard the teenager tell someone: "Even if the Prophet Muhammad comes down [from heaven], I will not become a believer."
The SOHR published a photograph of what it said was the boy's face, which bore gunshot wounds to the mouth and neck. Its director, Rami Abdul Rahman, said it could not "ignore these crimes, which only serve the enemies of the revolution and the enemies of humanity".
The main opposition body, the Syrian National Coalition, said that if the reports were true, it "would constitute a crime against humanity and those responsible must be brought to justice".
"The Syrian Coalition expects those taking part in the revolution to abide by the ideals and principles of international covenants and treaties," it said in a statement.
The explosions happened at a police building in the busy commercial district.
The attack comes as regime forces prepare an assault to recapture the northern city of Aleppo after having retaken Qusair from the rebels.
Images on Syria's al-Ikhbariya TV showed a scene of widespread damage, with shop fronts blown out and debris littering the street.
Blood stains marked the pavement, while people milled around among broken glass and wreckage.
One bomber reached the first floor of the police station and detonated his explosives when he came under fire, police told the BBC's Lyse Doucet in Damascus.
Several walls collapsed from the force of the blast. Others are splattered with blood and flesh, and punctured by holes from ball bearings packed inside the bomb.
The entire floor is littered with broken furniture and chunks of plaster, police sources say.
Police say a second suicide bomber exploded his device shortly afterwards, on the street below, as police were leaving the station.
It is unclear whether police are among the dead and injured.
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The BBC's Jim Muir says it is not yet known who carried out the attacks
But residents of the city centre say the situation there is much quieter now than it was a few months ago, when rebel forces were pressing in around the city and a final assault seemed imminent.
Since then, a major counteroffensive by regime forces has pushed the rebels back from the nearby suburbs, especially on the eastern and southern flanks of the capital.
Syria's conflict started more than two years ago, with largely peaceful protests against the regime of President Bashar al-Assad.
But it has now turned into a civil war that has killed more than 80,000 people, according to the United Nations.
Sights on Aleppo Syrian government forces are reported to be preparing for a major offensive on rebel-held parts of Syria's largest city, Aleppo.
Opposition activists in Aleppo told the BBC that military reinforcements - including fighters from Hezbollah, the Lebanese Shia Islamist group - had already been sent to parts of the city.
Aleppo has been carved up into areas controlled by rebels and the government
Much of northern Syria has been controlled by rebel groups since last year, and the front lines inside Aleppo have been largely static for months.
But the fall of Qusair last Wednesday has been seized on by the military leadership in Damascus as a decisive victory.
In view of the Syrian regime's advance, Washington could decide this week to start arming the rebels, US officials say.
A State Department spokeswoman said on Monday that the US would "continue to look for ways to help the opposition and increase aid... The president has talked about how boots on the ground is not an option - so all options short of that".
Boy's 'execution'
In a separate development on Sunday, the Syrian Observatory for Human Rights (SOHR) reported that Islamist rebels in Aleppo had executed a 15-year-old boy in front of his parents as punishment for what they regarded as a blasphemous comment.
The UK-based activist group said Mohammed Qataa was shot in the face and neck a day after being seized by the rebels, who allegedly overheard the teenager tell someone: "Even if the Prophet Muhammad comes down [from heaven], I will not become a believer."
The SOHR published a photograph of what it said was the boy's face, which bore gunshot wounds to the mouth and neck. Its director, Rami Abdul Rahman, said it could not "ignore these crimes, which only serve the enemies of the revolution and the enemies of humanity".
The main opposition body, the Syrian National Coalition, said that if the reports were true, it "would constitute a crime against humanity and those responsible must be brought to justice".
"The Syrian Coalition expects those taking part in the revolution to abide by the ideals and principles of international covenants and treaties," it said in a statement.