Syrian state media said only that its air defences intercepted missiles over the capital overnight while Israel did not immediately comment on the strikes.
Israel routinely fires missiles at what it says are Iranian targets in Syria, where elite Iranian forces and allied militia play a key role.
According to the Syrian Observatory for Human Rights, the strikes launched late on Thursday hit military targets in the area of the international airport.
Rami Abdel Rahman, the director of the Britain-based Observatory, said the dead were three Syrian soldiers and four members of Iran's Revolutionary Guard.
A Syrian army source quoted by SANA said the attack took place at 11:45 pm (2145 GMT), when an AFP correspondent in Damascus heard loud blasts.
"Our air defences intercepted hostile targets over the skies of Damascus," SANA said.
It said the "missiles were launched from over the occupied Golan Heights".
Several missiles were intercepted before they could reach their targets, said the agency, which rarely admits to casualties among army or Iranian ranks.
The Observatory said the targets in the airport area, which has been repeatedly hit in similar strikes, included an arms depot.
Arms delivery?
The strikes on the airport compound came just after the arrival of a cargo plane, according to Abdel Rahman, who could not specify where the aircraft had flown in from.
He said a vehicle had also been hit on the road towards central Damascus in what seems to have been an attempt to stop a weapons delivery.
SANA released a video of what appeared to be Syrian air defences firing over Damascus.
Israel's political leadership has spoken publicly of the bombing campaign, although the army rarely comments on individual strikes.
Israel argues that Iran's presence across several parts of Syria, including near the Golan Heights it occupies, is a threat and that it will continue its strikes.
On February 6, Israeli air strikes killed 23 Syrian and foreign fighters in Syria.
Among them were three Iranians as well as 12 Syrian and foreign members of Tehran-backed militias.
Iran has deployed its Islamic Revolution Guard Corps (IRGC) in Syria to support President Bashar al-Assad, who clung on by a thread earlier in the nine-year-old conflict.
It also holds sway over a myriad militia groups whose ranks include fighters from Afghanistan, Iraq, Lebanon and other countries.
The head of the IRGC's elite Quds Force, Qassem Soleimani, was killed in a US drone strike on Baghdad airport at the beginning of this year.
Soleimani was the mastermind of Iran's foreign operations but Iran's forces remain heavily involved in military operations carried out in Syria.
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