"This morning, Israeli warplanes fired several missiles from Lebanese airspace on one of our military positions in the Damascus countryside," said an army statement carried by state media.
"Our air defence systems blocked them and destroyed most of them."
An AFP correspondent in Damascus heard loud blasts around 3:30 am (0130 GMT).
The Syrian Observatory for Human Rights, a Britain-based monitor of the war, said at least some Israeli missiles had hit military targets near Damascus.
"Syria's air defence system blocked some of the missiles, but others hit ammunition depots near Jamraya," Observatory head Rami Abdel Rahman said.
Jamraya, which lies just over 10 kilometres (seven miles) northwest of Damascus, is home to several military positions and a branch of the Syrian Scientific Studies and Research Center (SSRC).
The US has repeatedly imposed sanctions on the SSRC for its alleged role in chemical weapons production. France has also imposed sanctions on the agency.
An Israeli air strike hit the facility in May 2013 and another hit an SSRC branch in western Syria in September.
Israel has carried out dozens of air strikes on the Syrian armed forces and their allies since the civil war broke out there in 2011.
It says it is aiming to stop advanced weapons deliveries to Hezbollah, a Lebanese armed group allied to the Syrian government with which Israel fought a month-long war in 2006.
An Israeli army spokeswoman had no immediate comment.
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