Russia admitted the plane had entered Turkey “by mistake” and assured Ankara it would not happen again, he said.
But a senior US official said the Obama administration does not believe the incursion was an accident, and officials are in urgent talks with allies about what to do.
In Madrid, US defence secretary Ash Carter said that the US is conferring with Turkish leaders about the infringement.
The incident comes amid Turkish concerns over Russian air strikes in Syria that have targeted some foreign-backed insurgents. Turkey and Russia also have conflicting positions on the Syrian regime, with Russia backing president Bashar Assad and Turkey insisting on his removal.
Mr Davutoglu said during an interview with Haber Turk television that Nato member Turkey would enforce its rules of engagement if its airspace is violated. Those rules call for the treatment of any element approaching the Turkish border from Syria as an enemy.
“The Turkish armed forces have their orders,” he said. “The necessary will be done even if it’s a bird that violates Turkey’s border ... Our rules of engagement are clear.”
A foreign ministry statement said that a Russian warplane entered Turkey’s airspace near the town of Yayladagi, in Hatay province, on Saturday. Two F-16 jets intercepted the Russian aircraft and forced it to fly back into Syrian airspace.
Also on Monday, Turkey’s military said a MIG-29 jet had harassed two Turkish F-16s for five minutes and 40 seconds on Sunday by locking its radar on to them.
In a brief statement, the military said the incident occurred while 10 F-16s were patrolling the Turkish-Syrian border. The military said it did not know which country the MIG-29 belonged to.
Turkey summoned the Russian ambassador and demanded that Russia avoid future infringements, the foreign ministry statement said. It warned that Russia would be held “responsible for any undesired incident”, that may occur. The same message was also relayed to Russian foreign minister Sergey Lavrov by telephone.
Nato secretary general Jens Stoltenberg expressed solidarity with Turkey and said the situation would be taken up at a meeting later.
“I call on Russia to fully respect Nato airspace and to avoid escalating tensions with the alliance,” Mr Stoltenberg said. “I urge Russia to take the necessary steps to align its efforts with those of the international community in the fight against Isil.”
Mr Davutoglu told Haber Turk television that Russia assured Turkey that the airspace would not be violated again.
“The information we got from Russia this morning is that it was an incident that occurred by mistake,” he said. “They said they are respectful of Turkey’s borders and that it would not happen again.”
Last week, Turkey issued a joint statement with its allies involved in the US-backed campaign against Islamic State asking Moscow to cease attacks on the Syrian opposition and to focus on fighting IS.
On Sunday, President Recep Tayyip Erdogan said the Russian air strikes were unacceptable and a grave mistake that could alienate Moscow in the region.
Russia says the air strikes that began on Wednesday are targeting IS and al Qaida’s Syrian affiliate, but at least some of the strikes appear to have hit Western-backed rebel factions.