A spokesman said the plane, an F-4 Phantom, was dealt with "according to the laws that govern such situations", the state news agency Sana said.
The Turkish prime minister said his country would "take the necessary steps" once all the facts were known.
The Turkish and Syrian navies are meanwhile engaged in a joint search for the two missing crew members.
The F-4 Phantom disappeared over the Mediterranean, south-west of Turkey's Hatay province, near the Syrian coast.
The Turkish military said it lost radio contact with the F-4 at 11:58 (08:58 GMT) while it was flying over Hatay, about 90 minutes after it took off from Erhac airbase in the province of Malatya, to the north-west.
A Syrian military statement said that an "unidentified air target" had penetrated Syrian airspace from the west at 11:40 local time (08:40 GMT), travelling at very low altitude and at high speed.
It said that in line with the laws prevailing in such cases, Syrian air defences engaged the craft, and scored a direct hit about 1km (0.6 miles) from its coastline.
It burst into flames, and crashed into the sea at a point 10km (6 miles) from the village of Om al-Tuyour, off the coast of Latakia province, well within Syrian territorial waters, the statement added.
Syrian television showed a map charting the aircraft's movements, coming in from over the sea near northern Cyprus.
The statement said that after it "became clear the target was a Turkish military plane which had entered our airspace", the naval commands of the two countries were in touch, and a joint operation was going on to find the missing crew members.
Earlier on Friday evening, Turkish Prime Minister Recep Tayyip Erdogan held a two-hour emergency meeting with his interior, defence and foreign ministers and the Chief of the General Staff, Gen Necdet Ozel.
"As a result of information obtained from the evaluation of our concerned institutions and from within the joint search and rescue operations with Syria, it is understood that our plane was brought down by Syria," Mr Erdogan's office said in a statement afterwards.
"Turkey will present its final stance after the incident has been fully brought to light and decisively take the necessary steps."
A spokesman for the UN Secretary General, Ban Ki-moon, said he was following the situation closely.
"He hopes this serious incident can be handled with restraint by both sides through diplomatic channels," Martin Nesirky told reporters.
Given the breakdown in relations between the two countries over the Syrian conflict, this incident has the potential to provoke a serious crisis, the BBC's Jonathan Head in Istanbul reports.
Much will depend on whether or not the Turkish pilots have survived, our correspondent says.
If not, public anger might push the government into some kind of punitive action against Syria, he adds.
Relations between Nato-member Turkey and Syria, once close allies, have deteriorated sharply since the uprising against Syrian President Bashar al-Assad began in March 2011.
Tens of thousands of Syrian refugees have fled the violence across the border into Turkey.
Inside Syria, the violence continued on Thursday with state media reporting that "armed terrorist groups" had abducted and massacred 25 villagers in Aleppo province.
Activists said that rebels had shot dead 26 government supporters who were believed to be militiamen.
In Aleppo city, activists said a number of people died when security forces opened fire on a demonstration after Friday prayers.
Meanwhile, international envoy Kofi Annan has said it is time for the world to exert greater pressure to help bring the violence in Syria to an end.
Mr Annan called for Iran to be involved in attempts to end the violence, a proposal put forward by Russia but rejected by the US.
In a separate development, the BBC has learned that UK government officials have decided to prevent the head of the Syrian Olympic Committee, Gen Mowaffak Joumaa, from travelling to London for the Games.
The visa ban is believed to be linked to his relationship to President Bashar al-Assad's government.