The plane crashed in the Mediterranean, off the coast of the southern province of Hatay. Its pilots are still missing.
Mr Erdogan spoke of Turkey's "rage" at the incident and described Syria as a "clear and present threat".
Nato condemned the attack and voiced strong support for Turkey, after Ankara invoked Article 4 of Nato's founding treaty, which entitles any member state to ask for consultations if it believes its security is threatened.
Four of the six jets were scrambled on Saturday from the airbase of Incirlik in response to two occasions of Syrian helicopters flying close to Hatay province, Sunday's army statement said.
Â
Later, two more F-16s took off from a base near Batman, in southeastern Turkey, after Syrian helicopters were spotted close to the province of Mardin, it added.
The military said the helicopters flew as close as 6.5km (4 miles) to the border, according to the AP news agency.
Regional impact
The border incident comes after the UN and Arab League envoy on Syria, Kofi Annan, warned of the danger of the Syria conflict spilling over into the wider region if the bloodshed is not stopped.
He was addressing an international meeting of major international and regional powers in Geneva on Saturday, aimed at reviving the six-point peace plan for Syria brokered by Mr Annan.
The countries present at the Geneva talks reached an agreement calling for a ceasefire and a transitional government in Syria.
Western demands to exclude President Bashar al-Assad and his allies from the interim administration reportedly foundered on opposition from Russia.
