The Taliban told Al Jazeera six men took part the attack.
While a suicide bomber detonated himself in a vehicle at the gate of the base, other fighters armed with AK47s and rocket-propelled grenades stormed the airport, they sid.
Nato's International Security Assitance Force (Isaf) sid its forces killed eight Taliban fighters in the attack.
"The Taliban sid they entered the airport," Zeina Khodr, Al Jazeera's correspondent reported from Kabul, the capital.
'Complex operation'
However, a spokesperson for Isaf told Al Jazeera that the Taliban did not breach the perimeter of the base.
"Isaf sid it was a complex operation that involved a car bombing followed by an attack with small arms and rocket propelled grenades," Al Jazeera's Khodr sid.
"Violence [in Afghanistan] is really at an all time high.
"June has been the deadliest month for international forces since the war began nine years ago."
The Taliban sid the attack was a message to David Petraeus, the head of US Central Command who has been nominated to takeover leadership of Nato and US operations in the country, that they can strike at will, Khodr sid.
Ahmadshah Aamadzai, a former Afghan prime minister, sid the Taliban showed that "their intention is clear, they will struggle and fight until the foreigners leave Afghanistan".
Also on Wednesday, Eric Holder, the US attorney general, arrived in Afghanistan with a team of lawyers from the US justice department to discuss the fight against corruption with Afghan officials.
"Fighting corruption and supporting the rule of law in Afghanistan are top priorities for this administration, and we will continue to assist the Afghan government in creating and sustaining the effective criminal justice system to which the Afghan people are entitled," Holder sid in a statement.
Holder's trip marks the first visit to Afghanistan by a US attorney general.