Anadolu reported that the restaurant's owner told local media that two armed men wearing "civilian clothes" had aimed "directly" at consular staff dining at HuQQabaz, a Turkish restaurant with branches in Turkey, Kuwait and Germany.
Turkey's Foreign Ministry said one Turkish consulate employee was killed in the attack.
It is unknown who carried out the attack.
The Iraqi Foreign Ministry also condemned the killing and said that it was working closely with local authorities in Erbil to find the perpetrators.
Kurdish authorities sealed off roads around the scene and searched nearby cars and businesses.
Two Iraqi nationals were also shot in the restaurant of which one was killed and the other was wounded, Iraq's Foreign Ministry spokesman Ahmed Sahhaf said.
No-one has so far claimed the attack but Turkish Kurdish militants are based in the region.
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Turkey is waging a ground offensive and bombing campaign in Iraq's mountainous northern region to root out pockets of the Kurdistan Workers' Party (PKK).
Initial reports said that three Turkish diplomats were killed.
Secretary of State Mike Pompeo condemned the attack "in the strongest possible terms".
The PKK is classified as a "terrorist" group by Turkey, the United States and the European Union because of the three-decade insurgency it has waged against the Turkish state.
The PKK meanwhile confirmed that one of its senior leaders - Diyar Gharib Mohammed, who sat on the presidential council of an umbrella organisation to which the PKK belongs - had been killed in a Turkish air strike.
In June, a group called the Self-Defence Forces of Southern Kurdistan was established in the region to attack Turkish military outposts on Iraqi territory and remove "occupiers".