Syria's state-run Al-Ikhbariya TV was broadcasting footage from the fence of the United Nations buffer zone between Syrian and Israeli forces minutes before the plane was targeted.
The amount of territory in Syria controlled by the Islamic State has been declining.
The aircraft were dropping bombs, apparently as part of a Russian-backed Syrian government push into areas previously held by anti-government forces.
The military says that there had been an increase in the internal fighting in Syria since the morning hours, including an increase in the activity of the Syrian Air Force.
"It was shot down and it crashed", Israeli military spokesman Lieutenant-Colonel Jonathan Conricus told reporters. Israel shot down the drone, and an Israeli warplane was shot down in the course of the aerial attacks that followed.
The Syrian jet pilot may have strayed into Israeli territory by accident, but Israel's UN Ambassador Danny Danon explained why Israel can't take any chances.
The jet was either a Sukhoi 22 or Sukhoi 24, the IDF said Tuesday.
Iran dismisses Trump's explosive threat to country's leader
On Sunday night at half past 11 o'clock, President Trump sent a public message to Hassan Rouhani , his Iranian counterpart. But he said the Iranian "people and the armed forces will stand up against our enemies and will not come up short".
Tuesday's incident comes just a day after Israel deployed its David's Sling missile defence system - designed for use against medium-range missiles - for the first time.
It was Israel's first operational use of the mid-range David's Sling, which is jointly manufactured by USA firm Raytheon Co.
Photos and video of the smoke trail left by the Patriot missile interception have surfaced on social media.
It is only the second such incident along the border in 30 years. Both pilots ejected and were injured, one critically.
Israel said it shot down the jet after it flew into Israeli territory.
'The IDF monitored the advance of the fighter jet, which infiltrated about 2km (1.25 miles) into Israeli airspace. The pilot's fate is not clear.
Thousands of Syrians, fleeing the fighting, established tent camps along the border.
But with President Assad's men closing in on jihadist groups on the Syrian side of the zone - and pictured reportedly hundreds of metres from an Israeli observation post - Israel has been ratcheting up its rhetoric.
"The mere fact that those would-be humanitarian activists, pretending to work for the sake of the Syrian people, had to be "rescued" from their own country, notably following an amnesty declared by the Syrian Government, suggests that the White Helmets" had reasons to fear that their crimes might be furthermore exposed", the statement reads. An worldwide effort, including the United States and Canada, backed the evacuation attempt, which saw the civilians transferred through Israel to Jordan.