The military base at the airport has been sealed off, but air operations that were immediately suspended resumed within an hour, Colonel Koziel Bart, a Nato officer, said.
The Taliban quickly claimed responsibility for the attack.
Witnesses described smoke rising in the distance at the main military entrance to the airport.
"I saw three or four Landcruisers for the foreigners just in front of the gate ... then there was a car and it hit them then blew up," a shopkeeper named as Humayun told Associated Press news agency.
The airport is used for both civilian and military purposes. It also serves as a base for foreign troops fighting the Taliban.
"The airport is an important target for the Taliban, coming just three weeks after they targeted the main Nato headquarters here in Kabul"
James Bays, Al Jazeera correspondent
The airport is also set to become a new headquarters for Nato forces in Afghanistan.
"Arguably this [the airport] is the second most important strategic base in the capital city, not just because it is an airport but also because they were changing the command structure of the Nato force here," James Bays, Al Jazeera's correspondent in Kabul, reported.
The US is introducing General David Rodriguez, previously the commander of the 82nd Airborne Division in eastern Afghanistan, to run day-to-day operations by foreign forces, while General Stanley McChrystal, the US top military commander in Afghanistan, will focus on strategy.
Bays said: "It [the airport] is an important target for the Taliban, coming just three weeks after they targeted the main Nato headquarters here in Kabul."
Seven civilians were killed in a suicide attack on the Kabul headquarters of the International Security Assistance Force (Isaf) on August 15. The attack has been blamed on the Taliban.
Four American and 10 Afghan soldiers were killed on Tuesday when Afghan and international forces confronted Taliban fighters in the eastern Afghan province of Kunar, local authorities and the US military said.
"Fighting is continuing" in the district of Sirkanay, Hamisha Gulab Shinwari, the head district official, told AFP news agency.
The loss of the four American soldiers was confirmed by a US army spokesman in Kunar.
Located on the border with Pakistan's tribal areas, the mountainous province of Kunar has been the most prone to armed groups’ attacks among Afghanistan's eastern provinces.
The death of the four Americans brings to 329 the number of foreign soldiers killed in Afghanistan this year. Of that number, 190 were American.