The interior ministry said a group of insurgents had opened fire with automatic weapons and rocket-propelled grenades after seizing a building under construction at the airport.
The attack began around 4:30 am (0000 GMT) but no casualties have been reported so far, the ministry said.
"The situation will be brought under control before long," General Ayub Salangi, the deputy interior minister, told AFP as security forces armed with automatic rifles took up position.
Civilian flights from the international airport north of Kabul have been suspended, another Afghan official said.
The airport lies next to a sprawling military base run by the NATO-led International Security Assistance Force, whose troops are withdrawing from Afghanistan after more than a decade of war, as the country lies in the grip of a power struggle following disputed presidential elections.
ISAF and Afghan military helicopters were seen hovering over the area during Thursday's attack, which came after a devastating suicide bombing at a busy market in southeastern Paktika province on Tuesday killed at least 42 people.
Pictures purporting to capture the impact of the attack on the surrounding area have surfaced on social media.
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