A roadside bomb blamed on the rebels, meanwhile, killed 14 police officers in eastern Turkey.
The troops crossed the border as part of a "hot pursuit" of rebels belonging to the Kurdistan Workers' Party, or PKK, who were involved in a roadside bomb attack that killed 16 soldiers on Sunday, the government official said.
"This is a short-term measure intended to prevent the terrorists' escape," the official said on condition of anonymity in line with government rules that bar officials from speaking to reporters without prior authorization.
The official didn't say how many troops crossed into Iraq, or exactly how long the operation would last.
Turkey's military has carried out numerous airstrikes and ground incursions into Iraqi territory in pursuit of the PKK over the past decades, but Tuesday's invasion was the first since 2011 and comes amid a sharp spike in violence between the PKK and the security forces.
The private Dogan news agency, citing unnamed military sources, said two battalions from Turkey's special forces crossed the border. A spokesman for PKK, Bakhtyar Dogan, told The Associated Press that the Turkish forces entered from Zagros mountains area in Iraqi Dahouk province. Dogan, the PKK spokesman, said no clashes had erupted.
Earlier, Kurdish rebels were suspected of detonating a bomb in the eastern province of Igdir as a police vehicle escorting a group of customs officials to a border gate was passing by, the Anadolu Agency reported. Fourteen police officers were killed while others were wounded in the attack in the province that borders Armenia, the agency said.
The customs officials were going to a border crossing between Turkey and the Azerbaijani enclave of Nakhchivan following the kidnapping of a group of customs officials by Kurdish rebels. The officials were released unharmed on Tuesday.
The 16 soldiers were killed in a similar attack by the PKK on Sunday. It was the deadliest assault on Turkish security forces since the renewed fighting erupted in July, derailing a peace process with the Kurds.
Earlier, Turkey's state-run news agency said Turkish jets had carried out airstrikes against a group of Kurdish rebels as well as the militants' camps in northern Iraq. It said up to 40 rebels died in the overnight aerial operations.
The airstrikes were confirmed by Firat news, a website close to the rebels.
Anadolu Agency said F-16 and F-4 fighter jets bombed a group of up to 25 PKK rebels believed to have carried out the bombing with improvised explosive devices that killed the 16 soldiers near the border with Iraq on Sunday. The agency said the jets struck the rebels on their escape routes in Iraq and also targeted six rebel camps there.
Anadolu said 53 planes were involved in the strikes.
The agency didn't cite a source for report and there was no immediate official confirmation.
Protests denouncing the PKK erupted in several cities across Turkey on Monday soon after the military announced the 16 soldiers' deaths. The pro-Kurdish political party said several of its local branches were vandalized during the demonstrations.
More than 200 people have been killed in the renewed fighting between the PKK and the security forces since July, including around 100 soldiers and police officers.
It comes amid increased political uncertainty in Turkey. The country is holding a new election on Nov. 1 following the ruling party's failure to form a coalition government after an election in June.
The PKK has been fighting for autonomy in southeastern Turkey since 1984. It is considered a terrorist group by Turkey and its allies.