A spokesperson for the terror group said militants fired "six Katyusha rockets" at the airport on Monday morning.
US officials said that anti-missile defences intercepted as many as five rockets.
Isis's failed attack came just hours after US forces launched a drone strike targeting an “imminent Isis-K threat” near Kabul airport on Sunday.
Witnesses on the ground said the strikes wiped out two cars close to a building near the airport.
US military officials said the attack prevented another deadly suicide attack at the airport.
It took out a vehicle carrying at least one person associated with the Afghanistan ISIS branch, officials added.
An Afghan official said that three children had died in the attack.
Last week an Isis suicide bomber killed 13 US troops and as many as 170 people, Britons among them, after detonating a device at the airport.
Late on Friday evening, the US retaliated in a separate drone strike in eastern Afghanistan, saying it had killed two high-profile members of the group.
Speaking after Monday's failed rocket attack, Captain Bill Urban, spokesman for the US military's Central Command, said there were no US casualties.
He said US forces used a defensive weapon known by the acronym C-RAM – a Counter-Rocket, Artillery and Mortar System – in response to the attack.
It targeted the rockets in a whirling hail of ammunition, he said.
The system has a distinct, drill-like sound that echoed through the city at the time of the attack.
Capt Urban said the Kabul airfield remains operational as the evacuation continued.
Meanwhile, Ross Wilson, the charge d'affaires at the US Embassy in Kabul now working out of the airport, insisted that evacuations remain ongoing Monday.
He dismissed as false claims that American citizens have been turned away or were denied access to the Kabul airport by US Embassy staff or American troops.
"This is a high-risk operation. Claims that American citizens have been turned away or denied access to HKIA by Embassy staff or US Forces are false," he said in a message on Twitter, using the acronym for the Kabul airport. He did not elaborate.
Additional reporting by Associated Press
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