Attack on Friday, which left one security guard wounded, was described by Islamabad as an assassination attempt against its ambassador to Kabul.
A foreign member of the Daesh terror group has been arrested in connection with an attack last week on Pakistan's embassy in the Afghan capital, the Taliban's chief spokesman said.
The Taliban-led Afghan government spokesman Zabihullah Mujahid said on Monday that special forces had arrested the person responsible for the embassy attack.
"This person is a citizen of a foreign country and a member of Daesh," he said.
"The investigation revealed that this attack was jointly organised by Daesh and rebels. Some foreign malicious circles are behind the attack and the aim was to create distrust between the two brotherly countries."
Mujahid declined to say what nationality the suspect was, or if it was the same person other officials reported had been arrested hours after the incident.
Pakistan's foreign ministry said on Monday that the ambassador, Ubaid ur Rehman Nizamani, was now in Islamabad "for consultations", but said there were no plans to close the embassy or withdraw its staff.
A security guard was wounded by shots fired at the embassy in Friday's attack, which Islamabad described as an assassination attempt against the head of the mission.
No country officially recognises Afghanistan's Taliban government, but Pakistan kept its embassy open in Kabul, even as Taliban took over in August last year, and maintains a full diplomatic mission.
Complicated relations with Taliban
Pakistan has complicated relations with the Taliban, with Islamabad long accused of supporting Taliban, while backing the US-led invasion of Afghanistan that toppled the group following the 9/11 attacks.
Pakistan is home to more than a million Afghan refugees, and the porous border they share is frequently the scene of clashes.
The Taliban have banned public protests, but Afghans have staged small demonstrations against Pakistan following such incidents.
The Taliban are at pains to portray Afghanistan as safe for diplomats, but two Russian embassy staff members were killed in a suicide bombing outside the mission in September in another attack claimed by Daesh.
Last month a gunman shot dead a Pakistan security guard at the Chaman border crossing, leading to its closure for a week.
Since returning to power the Afghan Taliban have insisted they would not allow foreign armed groups to operate in the country.