Suicide bombers linked to Islamic State are believed to have detonated two devices near crowds desperately seeking to flee the country and the Taliban.
The terror group claimed responsibility for the attack, in which at least 143 Afghans were also wounded.
In a televised address to the nation, US President Joe Biden told those behind the airport attacks: “We will not forgive. We will not forget. We will hunt you down and make you pay.”
The bombings were followed by a gunfight outside the airport, the head of US Central Command general Frank McKenzie.
The US death toll rose from 12 to 13 and was likely to rise even higher, a US official told Reuters. More than a dozen were wounded.
No members of UK forces are believed to have been injured, the Ministry of Defence said.
Pentagon spokesman John Kirby said there were two blasts near the airport, which is at the centre of the US and UK evacuation effort.
“We can confirm that a number of US service members were killed in today’s complex attack at Kabul airport,” he said.
“A number of others are being treated for wounds. We also know a number of Afghans fell victim to this heinous attack.”
He added that one explosion took place “at or near the Baron Hotel”, while the other was a short distance away at Abbey Gate.
Boris Johnson held an emergency Cobra meeting on Thursday evening and vowed to continue the evacuation effort in Afghanistan.
Transport Secretary Grant Shapps issued a NOTAM (Notice to Aviation) advising airlines avoid Afghan airspace under 25,000ft following the explosions.
The Ministry of Defence tweeted: “There have been no reported UK military or UK Government casualties following the incidents in Kabul.
“UK forces are working closely with our partners to provide security and medical assistance.
Thousands of Afghans have been gathering at the airport for days trying to flee the country since the Taliban takeover earlier this month. Western nations had warned of a possible attack.
The bombings came shortly after shots were reportedly fired at an Italian military transport plane as it flew out of Kabul airport.
An Italian journalist traveling on the flight told Sky 24 TG that the plane had been carrying almost 100 Afghan civilians when it came under fire minutes after take off on Thursday.
“The pilot reacted promptly and implemented manoeuvres to avoid being hit within minutes of taking off from Kabul. There was a bit of panic,” said the journalist.
A source at Italy's Defence Ministry appeared to back up the initial report, adding the plane was not damaged in the incident.
However the agency later reported that Italian intelligence believed the shots were fired to disperse the crowd near the airport, citing a government source.
Italy is one of a number of countries looking to evacuate thousands of foreigners and Afghans following the collapse of a Western-backed government in Kabul and the arrival of Taliban forces into the city.
Pressure to complete the evacuations has intensified with a looming August 31 deadline for Western forces to get as many people as possible - and themselves - out of the country.
Canadian forces halted their evacuations of around 3,700 Canadian and Afghan citizens on Thursday, saying they had stayed as long as they could.
"We wish we could have stayed longer and rescued everyone," acting chief of the defence staff General Wayne Eyre told reporters.
In an alert on Wednesday night, the US Embassy in Kabul advised citizens to avoid travelling to the airport and said those already at the gates should leave immediately, citing unspecified "security threats".
Britain also told people to move away from the airport area. Armed forces minister James Heappey said intelligence about a possible suicide bomb attack by IS militants had become "much firmer".
"The threat is credible, it is imminent, it is lethal. We wouldn't be saying this if we weren't genuinely concerned about offering Islamic State a target that is just unimaginable," Mr Heappey said on Thursday morning.