It was not immediately clear how the victims died. A U.S. official said troops had fired in the air to deter people trying to force their way onto a military flight that was set to take U.S diplomats and embassy staff out of the fallen city.
One witness, waiting for a flight out for more than 20 hours, said it was unclear if the five had been shot or killed in a stampede. U.S. officials at the airport were not immediately available for a comment.
Three bodies could be seen on the ground near what appeared to be an airport side entrance, in video posted on social media. Reuters could not verify the footage. Another witness said he had also seen five bodies.
The chaos came as Taliban officials declared the war over and issued statements aimed at calming the panic that has been building in Kabul as the militants, who ruled from 1996 to 2001, routed the U.S.-backed government's forces.
President Ashraf Ghani fled from the country on Sunday as the Islamists entered Kabul virtually unopposed, saying he wanted to avoid bloodshed.
Taliban spokesperson Suhail Shaheen said in a message on Twitter their fighters were under strict orders not to harm anyone.
"Life, property and honour of none shall be harmed but must be protected by the mujahideen," he said.