The strike, near Kabul airport, killed 10 members of one family, including six children, relatives have told the BBC.
The US military said it was targeting a vehicle carrying at least one person associated with the Afghan branch of the Islamic State group.
The US was assessing and investigating reports of civilian deaths, it added.
American commanders said there were “significant secondary explosions” after the drone strike – implying there were explosives at the scene – which may have harmed people nearby.
The youngest child to be killed was two-year-old Sumaya, and the oldest child was 12-year-old Farzad, the family told the BBC.
“It’s wrong, it’s a brutal attack, and it’s happened based on wrong information,” Ramin Yousufi, a relative of the victims, said.
He added, tearfully: “Why have they killed our family? Our children? They are so burned out we cannot identify their bodies, their faces.”
Another relative, Emal Ahmadi, told the BBC that it was his two-year-old daughter who was killed in the strike.
Mr Ahmadi said he and others in the family had applied for evacuation to the US, and had been waiting for a phone call telling them to go to the airport.
That included one of his relatives, Ahmad Naser, who was killed in the strike and had previously worked as a translator with US forces. Other victims had previously worked for international organisations and held visas allowing them entry to the US.
Pentagon press secretary John Kirby said the US was “not in a position to dispute” the reports.
“Make no mistake, no military on the face of the Earth works harder to avoid civilian casualties than the United States’ military, and nobody wants to see innocent life taken,” he said.
“We take it very, very seriously and when we know that we have caused innocent life to be lost in the conduct of our operations, we’re transparent about it.”
He then defended the intelligence about “what we believed to be a very real, a very specific and a very imminent threat” against Kabul’s Hamad Karzai International airport from IS-K (Islamic State Khorasan Province), IS’s Afghan affiliate.
In an earlier statement, US Central Command said there had been a number of “substantial and powerful subsequent explosions” following the drone strike.
It said the explosions suggested there had been “a large amount of explosive material inside, that may have caused additional casualties”.
Analysis box by Lyse Doucet, chief international correspondent
The last day of August will mark the end of what some describe as the end of US-led military engagement in Afghanistan, but the Taliban will proclaim it as the end of foreign occupation. On 1 September, Afghans will wake up and wait to see what this new chapter brings.
With every chapter in this 40-year war, Afghans dared to hope that the next chapter would be better than the last. I can say that of all the chapters I’ve witnessed over the last three decades, this is the most uncertain chapter yet.
The Taliban are promising to rule for all Afghans. They’re promising to give Afghans the best education system in the world, to meet all of their needs. It’s a huge challenge to move from an insurgency to governing again. The world will be watching – but watching most closely of all will be Afghans, wondering and hoping against hope that they will have a life they can call their own, in a country they still feel a sense of belonging to.
Afghans tend to say they hold on to hope because it’s the last thing they lose.
For the thousands who left, some of the best and brightest of this generation, who were educated, trained and prepared during this window of space provided by 20 years of international engagement – they have now not just left their country, they have lost their country.
They have lost their dreams, their hopes, everything they built up in the past 20 years. And it will be so, so painful for such a long time to come.
The US has been on high alert since a suicide bomber killed more than 100 civilians and 13 US troops outside the airport last Thursday. IS-K claimed responsibility.
Many of those killed had been hoping to board one of the evacuation flights leaving the city, which fell to the Taliban on 15 August.
The US had repeatedly warned of an increase in attacks as 31 August – the date set for the Americans’ withdrawal from Afghanistan – drew closer.
On Monday, a US anti-missile system intercepted a rocket flying over the capital towards the airport, US officials said. IS-K claimed responsibility for a total of five rockets that were fired.
Video and pictures carried by local news outlets showed smoke wafting across the rooftops of Kabul, and what appeared to be a burning car on a street.