The condition of two other passengers was not immediately clear, they added. The passengers were reportedly Russian nationals.
The pilot of the plane was among the survivors, the Taliban’s chief spokesperson Zabiullah Mujahid said.
Two provincial officials in Badakhshan said two other passengers had died, and the other four survivors were being looked after by Taliban representatives who had reached the remote mountainous site of the crash.
Mr Mujahid said that the investigative team of the Taliban’s Islamic Emirate “continues their efforts to search for and provide assistance to the remaining individuals”.
On Sunday, the Russian aviation authority Rosaviatsia said the plane, which was registered in Russia and had six people on board, disappeared from their radar in the Afghan airspace on Saturday. The confirmation came shortly after the local police officials in Afghanistan said they had received reports of a crash.
The pilot had warned that the aircraft was running low on fuel just 25 minutes before the plane vanished from radar screens, adding that an attempt will be made to land the plane in the next available territory north of Afghanistan in Tajikistan, reported Russian news outlet SHOT citing an unnamed source.
According to the report, both engines of the plane had stopped working.
Rostavasia said that the plane, which was a charter ambulance, was coming from Thailand’s Utapao airport in Pattaya to Moscow using the Indian and Uzbekistan airspace.
The aircraft was a French-made Dassault Aviation Falcon 10 jet manufactured in 1978, Rosaviatsia said in a statement.
Officials at the Russian embassy in Bangkok said that the flight was carrying out a private medical evacuation from Thailand’s Pattaya.
"On board was a bedridden patient in serious condition, a Russian citizen, who was transported from one of the hospitals in Pattaya to Russia," the RIA news agency reported, citing a source at Thailand’s Utapao International Airport.
"She was accompanied by her husband, a private entrepreneur, also a Russian citizen, who paid for the flight."
A criminal case has been opened by Russia’s Investigative Committee to determine if safety rules had been violated.