Syrian state news agency Sana said its military air defences had thwarted "hostile targets" and shot down several of them, without elaborating.
Witnesses in Damascus said loud explosions rang out in the night sky.
"We have started striking Iranian Quds targets in Syrian territory. We warn the Syrian Armed Forces against attempting to harm Israeli forces or territory," Israel`s military said in a statement.
The overnight strikes followed cross-border attacks on Sunday in which Syria said it repelled an Israeli air attack. Israel said it intercepted a rocket fired at the Golan Heights.
"We have a permanent policy, to strike at the Iranian entrenchment in Syria and hurt whoever tries to hurt us," Israeli Prime Minister Benjamin Netanyahu said earlier.
The Israeli army also said a popular winter tourist site on Mount Hermon in the Israeli-controlled Golan Heights would be shut for the day. It added that otherwise things remained "routine" along the frontier with Syria.
Netanyahu last week acknowledged an Israeli attack on what he called an Iranian arms cache in Syria, where Tehran provides Damascus with vital support.
He told his cabinet Israel had carried out "hundreds" of attacks over the past years of Syria`s war to curtail Iran and its ally Lebanon`s Hezbollah.
Usually silent about its attacks on Iranian targets near its frontier, Israel has lifted the veil this month, a sign of confidence in a campaign waged amid occasional tensions with Syria`s big-power backer Russia.
(Uu.SYS/C/KR-AMS/C/F001) 21-01-2019 08:00:05