Details of the yesterday shooting are still unclear, with a spokesman for NATO's Resolute Support mission saying the incident is under investigation.
A US defence official in Washington told AFP that the American victims, who were employed under a US Defense Department contract to help train the Afghan air force, died from gunshot wounds.
Taliban spokesman Zabihullah Mujahid said a member of the insurgent movement was responsible for the attack.
"A brave Afghan mujahid infiltrator working in the military side of Kabul airport opened fire on invading American soldiers, killing three Americans," he said in a statement sent to media.
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NATO troops have adopted special security measures in recent years to try to counter the threat.
The airport in the Afghan capital is heavily guarded, with one section devoted to commercial aircraft and another area set aside for a NATO contingent.
Yesterday's attack came after a surge in Taliban violence over the past year.
At least nine people were killed earlier yesterday in the country's east when a suicide bomber struck at a funeral for victims of a roadside bomb attack.
The American soldiers, along with other NATO troops and private contractors, are helping the Afghans improve their logistics and build up a fledgling air force.