In the video, two masked gunmen speak to a man kneeling in front of them.
After killing him, the gunmen say they will kill others they are holding if the government does not agree to their demands. The demands are unspecified.
They identify themselves as belonging to the Islamic Movement of Uzbekistan, who are allied to the Taliban.
There had been speculation that the hostages, mostly from the Hazara ethnic minority, had been seized by Islamic State as they travelled from Iran to Kabul through Zabul province.
The Islamic Movement of Uzbekistan (IMU) was formed in the late 1990s and said it carried out the attack on Karachi airport in June 2014.
The BBC's David Loyn in Kabul says this is the first time they have issued a video showing a beheading.
In the footage, the gunmen ask the hostage what his job is, and he says that he is a soldier. Local news agencies report that the kidnappers want their comrades released from Afghan jails.
Along with other Taliban factions, the IMU was pushed out of its former stronghold in Pakistan's North Waziristan tribal region in an army offensive launched after the Karachi airport assault.
Unlike in Pakistan, Afghan Hazaras, who are mostly Shia Muslims, have been largely spared attacks by Sunni militants operating in both countries.
Militants who kidnapped 31 people from a bus in Afghanistan in February have released a video showing one of the hostages being beheaded.