The attack in Mezzeh is reported to have caused fires, but no casualties.
The Israeli military has made no comment, but it is the second time in a week that it is alleged to have carried out a strike on Syrian territory.
It is thought to have bombed weapons shipments intended for Lebanon's Hezbollah movement several times since Syria's civil war began in 2011.
Hezbollah, which fought a month-long war with Israel in 2006, has sent thousands of fighters to Syria to support President Bashar al-Assad's forces.
Profile: Lebanon's Hezbollah movement
Syria's state news agency, Sana, cited a military source as saying that a number of surface-to-surface missiles landed around Mezzeh airport at 03:00 (00:00 GMT).
They were reportedly fired by Israeli forces from a position "west of Tal Abu Nada", a mountain in the Israeli-occupied Golan Heights known as Mount Avital to Israelis.
The military source said the attack was part of Israel's "desperate attempts" to support "terrorist groups and raise their deteriorating morale", an apparent reference to recent losses on the battlefield by Western-backed rebel forces.
Israel, the source added, would be "held fully responsible for the repercussions".
The Syrian Observatory for Human Rights, a UK-based group that monitors the conflict in Syria, reported that explosions were heard in the vicinity of the airport overnight, and that fires had erupted inside the facility afterwards. It could not confirm the source of the blasts.
Last Wednesday, the Syrian authorities said Israeli jets had fired two air-to-surface missiles from Lebanese airspace towards Sabboura, a north-western suburb of Damascus that is located on the main road from the city to the Lebanese border.
The Israeli military also declined to comment on that incident.
Days earlier, it said the Israeli Air Force had killed four militants linked to so-called Islamic State in the Syrian-controlled Golan Heights, after they opened fire on a patrol on Israeli-occupied territory.
Syrian state media say an Israeli missile strike has targeted a military airport on the outskirts of Damascus.