"We will not let Azaz fall," Davutoglu was quoted as saying by private NTV television on his plane en route to Ukraine.
"The YPG (the People's Protection Units, a Syrian Kurdish militia) will not be able to cross to the west of the Euphrates (river) and east of Afrin," he added.
Turkish artillery has struck at Kurdish militia targets in Syria since the weekend, with Ankara insisting it was returning fire.
Turkey confirmed today that it shelled advancing Kurdish fighters in northern Syria for a third day.
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Foreign ministry spokesman Tanju Bilgic said Ankara "retaliated in kind" when an attack from an area under YPG control targeted a border post in Turkey.
The Damascus regime has condemned Ankara over the shelling while urging the United Nations to act.
"Currently YPG elements were forced out of the Azaz neighbourhood. If they come closer to Azaz, they will receive the most severe response," Davutoglu said.
Turkey has been gravely concerned by the moves of the Kurdish fighters on Azaz and has so far defied calls from its Western partners to stop artillery bombardments of YPG positions
The Syrian Democratic Forces (SDF), a US-backed Kurdish-Arab militia alliance in which the YPG plays a key role, has also seized the Minnigh air base in northern Syria.
Davutoglu warned the SDF to withdraw from the airport, issuing a veiled threat of possible Turkish military action if they failed to do so.
"They will withdraw from the airport... "If not, the airport will be rendered unusable," he said, without elaborating.
"Russia is using those tools to be able to corner Turkey. Therefore, our stance has a legitimate basis.