The 10 were killed when a car parked at their home was struck by an explosion on Sunday.
The US military said it was targeting a vehicle carrying at least one person associated with the Afghan branch of the Islamic State group.
It said people nearby may have been hit in the aftermath of the strike.
The youngest child to be killed was two-year-old Sumaya, and the oldest child was 12-year-old Farzad, the family said.
“It’s wrong, it’s a brutal attack, and it’s happened based on wrong information,” Ramin Yousufi, a relative of the victims, told the BBC.
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.
The US, Mr Ahmadi added, had made “a mistake, it was a big mistake”.
Pentagon press secretary John Kirby told reporters that they were “not in a position to dispute” reports of civilian casualties, and that they were “assessing and… investigating”.
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”.
Central Command had previously said the strike was successful at “eliminating an imminent” threat to Kabul’s airport.
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.
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.
The White House said President Joe Biden was briefed on the rocket attack.
“The President was informed that operations continue uninterrupted at HKIA (Kabul airport), and has reconfirmed his order that commanders redouble their efforts to prioritize doing whatever is necessary to protect our forces on the ground,” White House press secretary Jen Psaki said in a statement.
No US or Afghan casualties have been reported so far from Monday’s incident.
The United States has installed an anti-rocket and mortar system to protect the airport from further attacks.
Later on Monday, British ministers and officials will be taking part in a number of international meetings aimed at defining a joint approach to the Taliban in Afghanistan.
Foreign Secretary Dominic Raab will hold talks with his counterparts from the G7 group of wealthy countries, as well as Nato and Qatar.
At the United Nations in New York, France and Britain are expected to propose a Security Council resolution calling for a safe zone in Kabul to protect people trying to leave the country.