State TV quoted army sources saying that 14 others were hurt in the attacks, contradicting earlier official accounts that said no one had died.
The rebel Free Syrian Army (FSA) said it had carried out the attack, but did not mention suicide bombings.
State TV broadcast footage of a minibus slowing before exploding at the HQ.
Gunfire reverberated around the city for hours after the bombings, as rebels fought with soldiers at the compound.
Official media said “terrorist attackers” had opened fire inside the perimeter of the compound and in nearby streets, and security forces had confronted them.
State TV said those killed were guards at the compound, and both civilian and military personnel had been wounded.
Witnesses said the blasts ignited a fire that engulfed much of the main building that houses the army’s General Staff.
State media insisted earlier that no senior officers were hurt.
The blasts happened just before 07:00 local time close to one of the city’s busiest areas, Umayyad Square, which is dominated by government buildings.
Roads leading to the area were blocked off as the authorities dealt with the aftermath.
Diplomats living close to the area said the blasts were the biggest they had heard for months.
Buildings more than 1 km (half a mile) away shook violently under the force of the explosions.
The staff command compound represents the heart of the Syrian army.
And the attack comes days after FSA announced it had moved its command from Turkey to Syria in an apparent attempt to bolster its fight against regime forces.
Damascus resident Jean-Pierre Duthion said his entire building shook and windows were shattered.
“We all ran into the corridor, in the place without windows or anything, and we were just waiting until it stopped,” he said.
Y87SWJzUpmo
Image source: Wikimedia Commons President-elect Donald Trump celebrated his election victory at the Ultimate Fighting…
Millions of voters across the US chose to return Donald Trump to the White House…
Donald Trump declares victory in the US election as he addresses jubilant supporters in Florida.…
Stocks around the world are rising as Donald Trump appears to be on the cusp…
Donald Trump has won Pennsylvania, North Carolina and Georgia and taken a lead over Kamala…
Quincy Jones, the celebrated musician and producer who worked with Michael Jackson, Frank Sinatra, Ray…