An AFP correspondent said the blast caused extensive damage and that intense gunfire was heard shortly afterwards. The blast damaged the AFP Damascus office, blowing out the windows, but no staff were hurt.
State television broadcast scenes of devastation as huge plumes of thick black smoke billowed up around buildings, partly obscuring them.
Firefighters rushed to the area, attempting to control blazes started by the explosion which one state broadcaster said took place near a school, adding that children were believed to be among the dead and wounded.
The footage showed bloodied bodies with limbs askew and chunks of flesh strewn on the streets, with bystanders draping clothes or cardboard boxes over them.
One group of men worked to retrieve a body from a badly damaged yellow taxi, tugging at its jammed doors. A veiled woman wept as she walked from the scene, passing a man holding a terrified, sobbing young girl.
"I was in the street with my colleague when the ground shook beneath our feet," 32-year-old Anana told AFP, not far from Sabaa Bahrat Square.
The Syrian Observatory for Human Rights said 15 civilians and four soldiers were killed in the blast, which also wounded more than 60 other people. In all, at least 71 people died in violence across the country on Monday, the monitoring group said.
The attack near the Syrian central bank was not immediately claimed by any group.
"We say to those behind these attacks that the Syrian people... will move forward to crush these armed terrorist gangs," Prime Minister Wael al-Halaqi said, speaking to media at the scene.
On March 21, a huge explosion ripped through a Damascus mosque killing at least 49 people, including a key pro-regime Sunni cleric. And a month earlier, on February 21, at least 83 people were killed in a spate of bombings in the city.
The foreign ministry, meanwhile, said that Syria will not accept a chemical weapons team, as proposed by UN chief Ban Ki-moon, to probe the alleged use of chemical weapons.
Ban has "suggested a supplementary mission allowing the mission to deploy throughout Syrian territory, which is contrary to the demand Syria made to the United Nations," a ministry official said, cited by state news agency SANA.
He said "Syria can not accept such manoeuvres on the part of the UN secretariat general, bearing in mind the negative role that it played in Iraq and which cleared the way to the American invasion" of that country in 2003.
Syria had specifically requested "a neutral and honest technical team to visit the village of Khan al-Assal" in the northern province of Aleppo, the official said.
Ban said earlier on Monday that a UN inspection team was in Cyprus and ready to deploy to nearby Syria.
"The UN is now in the position to deploy in Syria -- in less than 24 hours all logistical arrangements will be in place," Ban said.
Syria asked for the investigation into its allegation that the opposition had used chemical weapons in Khan al-Assal on March 19. The rebels charge that it was government forces that deployed the munitions.
In Lebanon, a source close to the Shiite militant group Hezbollah said two of its members were killed fighting alongside government forces on Monday in the Qusayr area of central Syria near the Lebanese border.
Syria's conflict, now in its third year, is believed to have killed more than 70,000 people.