It also said that the bodies of two passengers killed in the accident were being moved to the Afghan capital from the remote crash site.
Russian aviation authorities said on Sunday the plane with six people thought to be on board disappeared from radar screens over Afghanistan on Saturday night and Afghan police said they had received reports of a crash in mountainous Badakhshan province.
"Four people from the crashed plane in Badakhshan were transferred to Kabul, the medical and rescue teams of the Ministry of Aviation and the Ministry of Defence have provided them with first aid," Taliban administration spokesman Zabihullah Mujahid said in a statement.
Video footage released by Mujahid's office showed the four men, some of whom had bruising visible on their faces and one with blood stains on his clothes, stepping off a helicopter with Taliban officials clad in winter jackets.
The video showed an unnamed Taliban official saying the health of the survivors was good.
"Alhamdulillah (praise be to God) last night we found that place (the crash site), a total of six people were in the plane, four of them are alive and two are dead," he said, adding that the bodies had been transferred to the northern provincial city of Fayzabad and were being brought to Kabul.
The flight that crashed had been carrying out a private medevac from Thailand's Pattaya, a popular tourist destination for Russians, to Moscow, Russian state-run TASS news agency reported, citing the Russian embassy in Bangkok.
About 25 minutes before the plane vanished from radar screens, the pilot warned that fuel was running low and that the plane would try to land at an airport in Tajikistan, Russian news outlet SHOT reported, citing an unnamed source.
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