"Reports of downing an Iranian drone flying over Israel and also Iran's involvement in attacking an Israeli jet are so ridiculous", state TV quoted Iran's foreign ministry spokesman Bahram Qasemi as saying.
"My impression is that it seems to be contained at this point", said a Western diplomat in the region.
The area was quiet by mid-afternoon.
"The Secretary-General stresses once again that all concerned in Syria and the region have a responsibility and must abide by global law and relevant Security Council resolutions".
But the Israeli military then said that the Syrians responded with substantial anti-aircraft fire.
Syrian media said its air defences had struck the jet.
Sources in Damascus said there were casualties among Syrian forces. They want an accommodation in Syria in which the regime, some of the rebels and Israel will somehow coexist.
The Israeli air force launched a further series of strikes targeting what it said were Iranian and Syrian military targets in Syria.
On Saturday night, after hours of consultations with the defense minister and military chief of staff, Prime Minister Benjamin Netanyahu described the Iranian drone flight as a "brazen" attempt to violate Israel's sovereignty, and said it was Israel's "right and duty" to respond.
Iranian and Iran-backed Shi'ite forces, including Hezbollah, have deployed widely in Syria in support of Assad.
In a statement published by Lebanon's NNA news agency, Hezbollah said the developments denote "the categorical fall of ancient equations". He added that there was no Iranian or Russian anti-aircraft fire.
Netanyahu, visiting the Golan on Tuesday, peered across the border into Syria and in public remarks warned Israel's enemies not to test its resolve. Gen. Tomer Bar, second in command of Israel's Air Force, noted that the operation was "the biggest and most significant attack the air force has conducted against Syrian air defenses" since the 1982 Lebanon War.
In a statement, Hezbollah said that the downing of the Israeli F-16 marked a "new strategic phase" in the conflict.
"I do not believe matters will develop to a regional war", the official said. On a visit to Israel last month, U.S. Vice President Mike Pence called Iran the world's "leading state sponsor of terror".
"The United States is deeply concerned about today's escalation of violence over Israel's border and strongly supports Israel's sovereign right to defend itself", said State Department spokeswoman Heather Nauert.
She said Washington continues to push back on the totality of Iran's "malign" activities in the region and called for an end to Iranian behavior that "threatens" peace and stability.
"We will not allow that regime to entrench itself militarily in Syria, as it seeks to do, for the express objective of eradicating our state", Prime Minister Binyamin Netanyahu said during a visit to Washington in December.
Israel remains technically at war with Syria and occupies a swathe 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 global community.
There has been an uneasy standoff since a ceasefire that followed a war in 1973, with United Nations observer forces manning a buffer zone between the two armies. "Iran seeks to use Syrian territory to attack Israel for its professed goal of destroying Israel".