The Taliban government said the Afghan delegation included the defence minister and the head of the national intelligence agency. A Pakistani delegation was due to depart on Saturday, the national broadcaster PTV had said a day earlier without giving further details.
Each country says it is responding to aggression from the other. Pakistan accuses Afghanistan of harbouring militants who carry out attacks in border areas, a charge rejected by the Taliban.
Regional powers including Saudi Arabia and Qatar call for calm as the violence threatens to further destabilise a region where groups including Islamic State and al-Qaeda are trying to resurface.
A 48-hour ceasefire intended to pause hostilities expired on Friday evening and hours later, Pakistan struck across the border.
Pakistani security officials confirmed to The Associated Press there were strikes on two districts in Afghanistan's eastern Paktika province.
The targets were hideouts of militant group Hafiz Gul Bahadur, according to the officials who spoke on condition of anonymity because they were not authorised to talk to the media.
One said the operation was a direct response to the suicide bombing of a security forces compound in Mir Ali, in Pakistan's Khyber Pakhtunkhwa province a day earlier.
The Pakistani Air Force raids killed dozens of armed fighters and there were no civilian deaths, they said.
But Afghan officials said the aerial assaults killed at least 10 civilians, including women and children. The attacks prompted the national cricket board to boycott an upcoming series in Pakistan.
Zabihullah Mujahid, the Taliban government's chief spokesman, condemned the "repeated crimes of Pakistani forces and the violation of Afghanistan's sovereignty".
Such acts were deemed provocative and viewed as "deliberate attempts" to prolong the conflict, he added.
The two countries share a 2600km border known as the Durand Line, but Afghanistan has never recognised it.
Pakistan is grappling with surging militancy, especially in areas bordering Afghanistan. It also accuses its nuclear-armed neighbour and rival India of backing armed groups, without providing any evidence.
Pakistan's army chief Asim Munir urged Afghans to choose "mutual security over perpetual violence and progress over hardline obscurantism".
"The Taliban must rein in the proxies who have sanctuaries in Afghanistan," he told an audience on Saturday at the Pakistan Military Academy in Kakul, Khyber Pakhtunkhwa.
