The attacks were in response to Syrian government forces shooting down an Israeli fighter jet on Saturday, and claims that an Iranian drone entered Israeli airspace.
The air attacks reportedly hit an airport on the outskirts of al-Suwayda, in southern Syria, and a weapons depot near the capital, Damascus.
Israel has sounded several warnings about the perceived, increased Iranian involvement along its borders with Syria and Lebanon.
Russia’s President Vladimir Putin has urged Netanyahu to avoid any steps that could escalate tension.
Netanyahu has held several consultations with Putin, who, for his part, has sent forces to back Syrian President Bashar al-Assad.
Following the Israeli attacks, the two spoke again on Saturday, with Netanyahu conveying Israel’s intention to counter Iran’s actions.
Russia‘s foreign ministry appeared to criticise Israel’s actions by calling for restraint and respecting Syria’s sovereignty.
“It is absolutely unacceptable to create threats to the lives and security of Russian servicemen who are in Syria at the invitation of its legitimate government,” it said.
However, Netanyahu told a cabinet meeting on Sunday that Israel would continue to strike against any aggression.
“Yesterday, we dealt severe blows to the Iranian and Syrian forces,” Netanyahu said.
“We made it unequivocally clear to everyone that our rules of action have not changed one bit. We will continue to strike at every attempt to strike at us. This has been our policy and it will remain our policy.”
Saturday’s air attacks marked the toughest Israeli aerial assault on Syria in decades.
“Our preliminary understanding is that the plane crashed due to a ground-to-air missile,” Lieutenant-Colonel Jonathan Cornicus, Israeli military spokesperson, said on Sunday.
“It was hit.”
The jet was one of eight planes carrying out the Israeli raid inside Syria, said the report.
In the first, direct public acknowledgment of an attack inside Syria during the country’s civil war, Israel also said it had bombed 12 military sites across the border – eight Syrian and four that it said were Iranian.
The Syrian military and its allies have denied that the Iranian drone violated Israel’s airspace, according to a report by Syrian state-run SANA news agency.
The joint operations room, run by Syria, Iran, Russia and the Lebanese armed group Hezbollah, said the drone was on a regular mission, gathering intelligence on fighters of the Islamic State of Iraq and the Levant (ISIL, also known as ISIS).
Sources in Damascus said there were casualties among Syrian forces.
The Britain-based Syrian Observatory for Human Rights (SOHR), which monitors the war in Syria through a network of activists on the ground, said that at least six Syrian troops and allied militia members were killed in the Israeli air attacks.
The six included Syrian and non-Syrian allied troops, the SOHR said, adding that Israel had targeted several bases east of the central province of Homs.
It said the bases are used by both Iranians and Russians deployed in support of the Assad government.
Israel is technically at war with Syria and occupies a chunk of the strategic Golan Heights that it seized in the Six Day War of 1967, and later annexed, in a move never recognised by the international community.
Against this backdrop, the UN chief has called “for an immediate and unconditional de-escalation of violence”.
“[All] concerned in Syria and the region have a responsibility and must abide by international law and relevant Security Council resolutions,” Antonio Guterres said in a statement on Saturday.
Stephane Dujarric, the UN chief’s spokesman, said: “The secretary-general is following closely the alarming military escalation throughout Syria and the dangerous spillover across its borders.”
The White House issued a statement in which it said it supports Israel’s “right to defend itself from the Iranian-backed Syrian and militia forces in southern Syria.
“We call on Iran and its allies to cease provocative actions and work toward regional peace,” it said.