Local officials said at least 16 people were injured in the blast.
The incident is the latest in a series of recent deadly attacks following the announcement of a new Taliban leader.
Talking after the attack, Afghan President Ashraf Ghani accused Pakistan of sending “messages of war” and harbouring terrorist training camps.
Kabul’s deputy police chief, Sayed Gul Agha Rouhani, told reporters that a suicide bomber had driven a vehicle into the first checkpoint on the road into the airport.
Images showed a large plume of smoke rising from the site of the blast and locals reported seeing ambulances approaching the scene.
A Taliban spokesman said the attack, which happened at about midday (07:30 GMT), targeted a convoy of foreigners.
But officials said the five people killed included four civilians and a border police officer.
Wahidullah Mayar, a spokesman for the public health ministry, said a young child was among those injured in the explosion.
The airport was closed for several hours.
‘We hoped for peace’
Pakistan has historically backed the Taliban and since coming to power last year, President Ghani has sought Islamabad’s support in his bid to kick-start peace talks with the militants.
But at a news conference on Monday, he attacked Pakistan.
“The last few days have shown that suicide bomber training camps and bomb-producing factories which are killing our people are as active as before in Pakistan,” Mr Ghani said.
“We hoped for peace but we are receiving messages of war from Pakistan. We can no longer see our people bleeding in a war that is exported from outside,” he added.
Monday’s Taliban attack came after three deadly attacks on Friday that killed at least 50 people.
A suicide bomber blew himself up near the city’s police academy on Friday evening, killing about 20 recruits.
A short while later, gunmen launched an attack on Camp Integrity, a Nato base that houses US special forces near the airport.
Eleven people, including an American soldier and eight contractors, were killed in the attack on the base.
Earlier on Friday, a truck carrying explosives was detonated near an army base in the Shah Shahid area of the capital, claiming 15 lives.
The Taliban said it was behind the attack on the police academy but did not mention the two other incidents.
“We suspect the upsurge in violence may be triggered by the succession battle within the Taliban,” Nicholas Haysom, the head of the UN mission in Afghanistan, told the BBC on Saturday.
Last week, the Taliban released a video in which they showed members of the group pledging allegiance to the new leader, Mullah Akhtar Mansour.
But analysts suggest there has been infighting within the group between supporters and opponents of Mansour.
The Taliban’s announcement that Mullah Omar had died led to a tentative peace process being suspended last week.