Yildirim told Turkish reporters in a briefing in Istanbul that the troops crossed into the YPG-controlled region in Syria at 0805 GMT from the Turkish village of Gulbaba, the Dogan news agency reported.
The state-run Anadolu news agency said that the Turkish army troops, whose number was not specified, were advancing alongside forces from the pro-Ankara rebel Free Syrian Army (FSA).
The Dogan news agency published images of tanks and armoured vehicles pushing through grassy terrain.
However this was the first confirmation that Turkish ground troops were now involved in the operation inside Syria.
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Yildirim was quoted as saying that the Turkish forces aimed to create a security zone some 30 kms deep inside Syria.
Turkey accuses the YPG of being the Syrian offshoot of the Kurdistan Workers' Party (PKK) which has waged a rebellion in the Turkish southeast for more than three decades and is regarded as a terror group by Ankara and its Western allies.