The spokesman for the former regime warned that another 60 bombers planning to attack foreigners had entered the capital and would attack International Security Assistance Force troops, coalition forces and people working for the United Nations and non-governmental organisations (NGOs).
"We claim the responsibility for the attack today in Kabul," the spokesman, who calls himself Abd al-Samad, told AFP by satellite phone.
"The attack was carried out by martyred Abd Allah, 35 years old and a citizen of Chechnya."
"He was walking toward the ISAF base in Kabul airport to carry out an attack. The second option was if he could not reach the base he would target ISAF patrols or other coalition vehicles driving on that road but he was arrested by Afghan security and he carried out the attack and killed himself and five others."
Samad claimed the bomber had accomplices who survived.
"A group of 60 bombers have entered Kabul city. Their target will be ISAF, the coalition, UN, and other NGOs or people working for foreigners," he added.
"We are more organised now; we will focus our attacks on cities now rather than the borders," he said, referring to a wave of violence along the eastern border blamed on Taliban fighters.
An Afghan intelligence chief was among the victims of the attack, about 200 metres from Kabul airport, where hundreds of International Security Assistance Force (ISAF) troops are based.
"Two of the dead bodies have military uniforms and two others are wearing civilian clothes," Kabul's deputy police chief Mutahullah Rahmani told news agencies. "We are investigating.
A spokesman for the 5,700-strong ISAF Force in Kabul said the blast appeared to have been caused by a car bomb rather than explosives wrapped around the bomber's body.
ISAF works alongside a 12,000 member US-led force trying to contain Al-Qaida and Taliban fighters.