Israel launched a series of predawn air strikes then followed up with artillery fire on an army base near the Syrian capital, Damascus, Syria's military said.
The allegation was not denied by Israeli Prime Minister Benjamin Netanyahu when questioned by reporters later on Tuesday.
Israeli jets fired missiles at the Qutayfeh area northeast of Damascus from inside Lebanese airspace at 2:40am local time (00:40 GMT), causing the Syrian army to retaliate and "hit one of its planes", the Syrian army said in a statement broadcast on state television.
UK-based war monitor, the Syrian Observatory for Human Rights, said the air strikes targeted Syrian army and Hezbollah weapons depots, igniting "successive explosions and fires, causing material damage".
After the air raids, Israel also launched rockets from the occupied Golan Heights, but the Syrian military intercepted them, the army statement said.
When asked about the alleged attacks inside Syria, Netanyahu said Israel was ready to stop weapons from reaching the Lebanon-based Hezbollah armed group, which has backed the Syrian government in its nearly seven-year civil war with rebel forces.
"We have a long-standing policy to prevent the transfer of game-changing weapons to Hezbollah in Syrian territory," the Israeli prime minister was quoted as saying by the Times of Israel.
"This policy has not changed. We back it up, if necessary, with action," Netanyahu added, without confirming or denying attacks were launched inside Syrian territory.
The report said the Israeli military wouldn't comment on the alleged attacks.
Syria wrote a letter to the UN Security Council demanding that it condemn Israel's cross-border strikes. Syria's foreign ministry also accused Israel of launching the attacks in support of opposition fighters.
"The recurring Israeli attacks on Syria will not succeed in protecting the terrorist organisations, which are Israel's partners and proxies," the official news agency SANA quoted the ministry as saying.
It said Israel was "primarily" assisting Islamic State of Iraq and the Levant (ISIL, also known as ISIS) and Jabhat Fateh al-Sham, a former al-Qaeda affiliate.
"The Syrian government once again warns Israel of the serious repercussions of its attacks on Syria and its support for armed terrorist organisations," the ministry said.
Meanwhile, SANA reported that rebel shelling of the capital killed five people and wounded 30 others on Tuesday.
It said 15 artillery shells hit the central, predominantly Christian neighbourhood of Bab Touma, adding Damascus has been targeted on a near-daily basis in recent weeks.
