"Whoever threatens to eliminate us, bears full responsibility", he said.
The military said the Mount Hermon ski site has been closed.
"The young people are impatient and fully ready to battle the Zionist regime and make them vanish from the Earth", Nasirzadeh said.
Monday's strikes were the second Israeli attack in less than 24 hours.
State news agency SANA said a "bomb blast" had hit southern Damascus "without leaving any victims".
Israel's Iron Dome, which has been operating since 2011, is a missile interceptor system that detects and shoots down incoming rockets.
The IDF provided extensive information on the attack, the first time it has done so since the IAF carried out strikes against Iranian targets in Syria in May 2018 and after the Iranians fired dozens of rockets at IDF targets in the Golan Heights.
The Israel Defence Forces (IDF) struck Iranian targets in and around Damascus, including the city's global airport, IDF spokesman Lt. Col. Jonathan Conricus said.
Netanyahu said that Israeli attacks mainly targeted Iranian positions while Syrian forces fighting alongside Iranians also came under fire, Reuters reported.
The announcement marks a rare departure from Israel's years-long policy of ambiguity regarding activities in neighbouring Syria.
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Also, Israeli military released footage showing the several Russian-made Pantsir-S1 (SA-22 according to the North Atlantic Treaty Organisation designation) missile defense systems took a direct hit during an Israeli airstrike in Syria.
The statement was issued hours after the purported rocket over the Golan Heights.
Syria said it had shot down several "hostile targets" without elaborating.
If Iran has it its way, Syria will be covered with Iranian missile bases, as well as cross-border attack bases for future raids against Israel by pro-Iranian forces, like Hezbollah and Shi'ite militias.
Throughout the war in neighboring Syria, Israel has regularly intervened to prevent Tehran's expansion in the beleaguered country.
Should the situation escalate, Israel could attack strategic targets in the Iranian homeland itself and not make due with defensive strikes in its own backyard. The London-based Palestinian newspaper Al-Quds Al-Arabi reported on Friday that Moscow informed the Israeli government that it intends to renovate Damascus Airport, and that further attacks would not be welcomed.
The targets included weapons depots belonging to the Lebanese Shiite movement Hezbollah and Iranian fighters, it added.
Earlier in the day, the Israel Air Force had conducted a series of airstrikes on Iranian targets in the war-torn country.