About 150 French soldiers who had been helping run the military airport handed over responsibility to a Turkish unit which will operate under the new NATO mission.
NATO's war in Afghanistan formally ended on Sunday, when the International Security Assistance Force (ISAF) was replaced by the US-led follow-up mission "Resolute Support".
France, which withdrew all its combat troops from the country two years ago, lost 89 soldiers and saw 700 injured in the war since 2001.
The foreign force will consist of the 12,500-strong NATO mission, most of them US troops, and a US counter-terrorism operation outside the NATO remit, though final numbers remain unclear.
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The Taliban issued another statement today celebrating the end of NATO's combat mission, adding that no peace talks could happen before all foreign troops leave.
"Today, they are evacuating their invading forces from Afghanistan while they are bitterly defeated by the just and pious Afghan resistance," the group said.
"The presence of foreign occupiers is main cause of instability and chaos."
The Taliban, who ruled Afghanistan from 1996 to 2001, have fought a resilient insurgency against NATO and Afghan forces for 13 years, with violence now at record levels nationwide.