UK foreign secretary Dominic Raab confirmed the deaths as he said a further two people were injured in the attack.
US officials now believe the terror incident was one blast at Kabul airport on Thursday, rather than the originally suspected two explosions.
Mr Raab said: “These were innocent people and it is a tragedy that, as they sought to bring their loved ones back to safety in the UK, they were murdered by cowardly terrorists.”
It is understood the child who died was a teenager, while those injured are an adult British national and an Afghan child with a British family.
The foreign secretary added: “Yesterday’s despicable attack underlines the dangers facing those in Afghanistan and reinforces why we are doing all we can to get people out. We are offering consular support to their families.
“We will not turn our backs on those who look to us in their hour of need, and we will never be cowed by terrorists.”
As well as the British casualties, officials have said at least 13 US troops and 60 Afghan nationals were killed – and more than 150 people were injured – in a “complex attack” on Thursday.
However, both Prime Minister Boris Johnson and US president Joe Biden have vowed to continue the evacuation effort in Afghanistan despite the attack.
SNP Westminster leader Ian Blackford said of the British casualties: “This is devastating news. My heart goes out to the victims, their families, our personnel on the ground, and to the vulnerable Afghan people stuck in this nightmare situation.
“The UK Government must do everything it can to ensure safe routes for the evacuation, relocation and resettlement of those left behind – and substantially increase the number of Afghan refugees the UK plans to take. We must not abandon those we have a responsibility to protect.”
The Pentagon on Friday said it had incorrectly reported there had been two bombings in Kabul.
Major General Hank Taylor of the Pentagon’s Joint Staff told reporters that it is now believed there was no attack at the Baron Hotel, where processing was taking place.
He said the US military report was incorrect and attributed the mistake to confusion in the aftermath of the violence.
Earlier UK defence secretary Ben Wallace warned the threat from terror groups will only “grow the closer we get to leaving” following the attack, believed to have been carried out by the IS-K affiliate of so-called Islamic State.
Despite airlifting nearly 14,000 people out of Afghanistan in the past two weeks, Mr Wallace said “the sad fact is not every single one will get out”.
He declined to give a timeline for the exit of British forces as they processed about a further 1,000 evacuees already in the airport, but he acknowledged it would come before the Americans withdraw, with Mr Biden having set the departure deadline of Tuesday.
Mr Johnson, appearing to point to the timetable set out by the Americans, said: “It’s certainly not something that … the timing of this is certainly not the one that this country would have chosen, and I think that everybody understands that.”
The Prime Minister added: “Of course, as we come down to the final hours of the operation there will sadly be people who haven’t got through, people who might qualify.
“What I would say to them is that we will shift heaven and earth to help them get out, we will do whatever we can in the second phase.”
Mr Wallace said the Baron Hotel processing centre was closed at 4:30am on Friday, as was the Abbey Gate to Hamid Karzai International Airport.
On Friday morning, the Ministry of Defence said 13,708 people had been evacuated by the UK from Kabul under Operation Pitting, which began on August 13.
Mr Wallace said this included nearly 8,000 Afghans eligible under the UK's relocation scheme for those who worked for the UK government and other vulnerable individuals, as well as 4,000 British passport holders.
“We will process the people that we’ve brought with us, the 1,000 people approximately in the airfield now, and we will seek a way to continue to find a few people in the crowds where we can, but overall the main processing is now closed and we have a matter of hours,” he said.
James Sunderland, MP for Bracknell and a former senior army officer, said: “The Prime Minister is between a rock and a hard place. He’s doing his best to honour UK commitments and sticking with the extant plan, but taking tactical advice on the threat.
“I’ve no doubt there is an imperative to bring UK nationals and UK troops home as soon as possible, but he’s rightly staying calm and doing his best for those on the list of entitled personnel, wanting to leave no-one behind and taking a calculated risk.”
Labour leader Sir Keir Starmer, who described the confirmed deaths of British nationals as "incredibly sad”, said the looming end of the evacuation from Kabul marked a “dark day”.
He said the UK Government had “serious questions to answer, adding: “The British Government must take its fair share of the responsibility ... despite having 18 months to prepare, their failure to plan and inability to influence others has contributed to this tragic political failure.
“We must urgently help the thousands who we have left behind, some of whom are eligible for relocation under the Arap scheme.”
Liberal Democrat leader Sir Ed Davey said a “safe passage corridor” must be created in the next 24 hours to help those who still wish to flee Afghanistan.
"The deaths of British people and many others in this appalling terrorist attack breaks our hearts,” he said.