The Syrian Observatory for Human Rights said there had been eight blasts around the base as missiles hit ammunition warehouses.
"In a desperate attempt to support terrorist organisations, Israeli enemy aircraft launched missiles from the north of Lake Tiberias (the Sea of Galilee) at 00:25 am (2225 GMT Thursday)," a military source told the state SANA news agency.
"The Syrian armed forces warns the Israeli enemy of the repercussions of this blatant aggression, and insists on continuing the war on terrorism to eliminate it," the source added.
The Israeli army had no comment on the strikes when contacted by AFP earlier on Friday.
An AFP correspondent heard several explosions and saw a large fire inside the Mazzeh airbase on the western outskirts of Damascus, with smoke visible across the capital.
The Syrian Observatory for Human Rights, a British-based monitoring group, said there had been eight blasts around the base as missiles hit ammunition warehouses.
Since the Syrian civil war erupted in 2011, the government has accused Israel of carrying multiple strikes on its territory, including in the Mazzeh area.
Several have targeted Lebanese militant group Hezbollah, an Israeli archfoe whose fighters have been supporting the Syrian army against rebels whom the government of President Bashar al-Assad regards as "terrorists."
Israel and Hezbollah fought a devastating war in 2006.