"Our air defences responded to an Israeli missile attack on Damascus international airport and shot down a number of hostile missiles," a military source said, quoted by SANA.
The agency, without giving any information on casualties or damage, posted footage and images of the air defences being activated.
In a shaky video, a small, bright explosion is seen in the night sky, with city lights in the distance.
AFP's correspondent in Damascus heard a loud blast late Saturday, followed by several smaller explosions.
The Israeli military declined to comment on the incident.
The Syrian Observatory for Human Rights, a Britain-based war monitor, said Saturday's strikes hit a weapons depot outside the airport.
"The missiles, suspected to be Israeli, destroyed an arms warehouse near the Damascus international airport," Observatory head Rami Abdel Rahman told AFP.
He had no immediate information on casualties.
Israel has vowed to prevent its arch-foe Iran, which is a main backer of Syria's government, from gaining a foothold in neighbouring war-torn Syria.
Earlier this month, Israel acknowledged having carried out more than 200 strikes in Syria over the past 18 months, mainly against Iranian targets.
It has also admitted to striking Syria to prevent what it says are deliveries of advanced weaponry to Lebanon's Hezbollah, an armed movement backed by Iran and which fights alongside Syrian troops.
The last reported Israeli strikes on Syria took place on September 4, when Syrian state media said the military's air defences downed several missiles in the coastal province of Tartus and in central Hama.
The Observatory also reported those raids and said they killed three Syrian soldiers.
Syria's conflict erupted in 2011 and has since killed more than 360,000 people, with millions more displaced internally and to neighbouring countries.
After losing swathes of territory to rebel groups, President Bashar al-Assad's troops have regained the upper hand and are now in control of around two-thirds of the ravaged country.
They were bolstered by nearly three years of air strikes by their key ally Russia and Iranian, Iraqi, Lebanese, and other foreign fighters on the ground.
Soldiers and other loyalist fighters had been amassing around Idlib, the largest rebel-held zone left in Syria, for several weeks.
But Russian Foreign Minister Sergei Lavrov denied that Syrian troops were planning a major offensive for the area.
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