US officials said the attack on Sunday was targeting a vehicle carrying large amounts of explosives and heading towards the city’s airport.
A military official said the strike caused “significant secondary explosions”.
Technical engineer Zemaray Ahmadi, 36, was killed with his sons Zamir, 20, Faisal, 16, and Farzad, 12, the family said.
It was also said six of his nieces and nephews were also said to have died including a boy and girl both aged two, girls aged five and seven, a six-year-old boy and a 28-year-old man.
A spokesman for America’s Central Command said in a statement: “We would be deeply saddened by any potential loss of innocent life.”
The area around the airport, the scene of frantic last-minute flights out of the city at the weekend, remains chaotic.
There have been reports of Taliban members hunting down people with links to the old regime who were unable to get out.
The UK pulled its final troops and diplomatic staff out on Saturday.
Foreign Secretary Dominic Raab will meet with international counterparts on Monday.
Top of the agenda is expected to be the Taliban’s promise to allow safe passage for foreign nationals and Afghans authorised to enter third countries.
The pledge is important amid fears the numbers of Afghans left behind who may be eligible for resettling was far higher than government estimates.
Shadow Foreign Secretary Lisa Nandy wrote to Mr Raab on Sunday to explain how Labour MPs alone had 5,000 potential cases in their inboxes.
She warned the government was working with a “serious underestimate”.