It is the first officially confirmed public execution since the hardline Islamist group seized power last year.
Reuters reports the execution in Farah province was of a man accused of stabbing another man to death in 2017, Taliban spokesperson Zabihullah Mujahid said, and was attended by senior Taliban officials.
The execution was carried out by the father of the victim, who shot the man three times, Mujahid said in a subsequent statement.
The case was investigated by three courts and authorised by the Taliban's supreme spiritual leader, who is based in the southern province of Kandahar, Mujahid said.
More than a dozen senior Taliban officials attended the execution, Mujahid said, including acting interior minister Sirajuddin Haqqani and acting deputy Prime Minister Abdul Ghani Baradar, as well as Afghanistan's chief justice, acting foreign minister, and acting education minister.
The United Nations mission in Afghanistan and a spokesperson for the UN Human Rights Office in Geneva issued statements condemning the execution and calling on the Taliban to establish an immediate moratorium on the death penalty.
UN human rights office spokesperson Jeremy Laurence said the death penalty was "incompatible with fundamental tenets of human rights, and its use cannot be reconciled with full respect for the right to life".
A Taliban spokesperson said in response to the United Nation's comments that retributive justice was the right of victims' family members.
"If they want to forgive, they can forgive. If they want to execute, they can execute ... retribution is a divine order and must be implemeted," said Taliban deputy spokesperson Yusuf Ahmadi.
Afghanistan's Supreme Court has announced public lashings of men and women accused of offences like robbery and adultery reported in several provinces recently, signalling a possible return to practices common under Taliban rule in the 1990s.
A spokesperson for the UN human rights office last month called on the Taliban authorities to immediately halt the use of public floggings in Afghanistan.
The Taliban's supreme spiritual leader met judges in November and said they should carry out punishments consistent with sharia (Islamic law), according to a court statement.
Public lashings and executions by stoning took place under the previous 1996-2001 rule of the Taliban.
Such punishments later became rare and were condemned by the foreign-backed Afghan governments that followed, though the death penalty remained legal in Afghanistan.
A member of the Taliban security forces stand guard at a checkpoint along a street in Jalalabad on 6 December, 2022. (file picture) Photo: AFP