It was not immediately clear who carried out the attack, or whether they had been apprehended.
North Atlantic Treaty Organisation did not release the identity of the American soldier killed or provide further details about where the incident took place.
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But a police officer said the shooting happened at the airport in Tarinkot, the capital of Uruzgan province, although they were unable to confirm this detail with any North Atlantic Treaty Organisation officials. On Twitter, the Taliban also gave Uruzgan as the location.
Pompeo also added, perhaps optimistically given Afghanistan's long history of resisting foreign forces, that the Taliban were "beginning to see that they can not wait us out".
Some of the USA forces are involved in counterterrorism operations, particularly against the local Islamic State group.
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Around 16,000 troops from US, NATO and other partner countries are deployed in Afghanistan.
A "conscientious" Afghan soldier, Ahmadi said, turned his gun on "American invaders" and killed four of them. The U.S. invaded Afghanistan following the September 11 terror attacks by al-Qaida, which the Taliban government at the time had harbored.
NATO's statement did not elaborate on the exact circumstances of the attack, but said it was under investigation.
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Insider attacks on Afghan forces - and deaths resulting from those attacks - have increased by more than 50 percent compared to the same time frame previous year. Afghan commandos carried out the operation with support from US special forces and air power. I.S.is also maintaining its eastern and northern footholds despite an intensified aerial bombing campaign by Afghan and USA forces.