According to Tasnim dispatches, five rockets hit the ammunition depot in the airport's compound early on Thursday, leaving fuel tanks burning there.
Two people were also reported injured as a result of the blast.
The report added that the blast was huge and could be heard in Damascus, particularly in southern areas.
Some reports suggest that the explosion was likely caused by an Israeli airstrike. The Tel Aviv regime has neither confirmed nor denied any role in the attack.
Israeli warplanes have targeted the Damascus airport and other bases around the capital in the past. The airport was hit by Israeli airstrikes in December 2014.
According to the report, Syrian warplanes are now flying over the area.
Meanwhile, the Arab country's military forces are gearing up to launch a fresh offensive against terrorists' positions in the districts of Jobar and Qaboon on the outskirts of the capital.
Over the past six years, Syria has been fighting foreign-sponsored militancy. UN Special Envoy for Syria Staffan de Mistura estimated in August that more than 400,000 people had been killed in the Syrian crisis until then. The UN has stopped its official casualty count in Syria, citing its inability to verify the figures it receives from various sources.