The footage shows her being shot several times in the back in Qol village in Parwan province just north of the capital Kabul - in front of a crowd of men.
The 22-year-old woman, named as Najiba, was married to a member of a hardline Taliban militant group and was accused of adultery with a Taliban commander, Parwan provincial spokeswoman Roshna Khalid said.
She said: "Within one hour they decided that she was guilty and sentenced her to death. They shot her in front of villagers in her village, Qol."
After the shooting, a villager handed the video over to the provincial government and "the security forces are preparing a big operation to find the culprits", she said, adding the execution happened late last month.
Public executions of alleged adulterers were common when the Taliban regime was in power from 1996 until 2001, when they were ousted by a US-led invasion for sheltering al Qaeda leader Osama bin Laden after the 9/11 attacks.
The Taliban have since waged an insurgency against the government of President Hamid Karzai, which is supported by 130,000 Nato troops.
The video begins with the woman, wrapped in a grey shawl, sitting at the edge of a ditch in a village surrounded by dozens of men, some perched on rooftops.
With her back to the crowd, a bearded man is seen reading verses from the Koran condemning adultery, before saying: "We cannot forgive her, God tells us to finish her. Juma Khan, her husband, has the right to kill her."
The video then shows a man in white being handed an AK-47 rifle.
He walks up to within a couple of metres of her, says 'Allahu akhbar' (God is great), aims and fires twice, missing each time. The third shot hits her in the back, she flings her arms wide and collapses.
He then fires another six shots into her body as the crowd cheers wildly, shouting "Long live Islam", "Long live mujahideen (holy warriors)". The gunman then fires four more shots into her body.
The Afghan government has said it "strongly condemns this un-Islamic and inhuman action by those professional killers and has ordered the Parwan police to find the culprits and bring them to justice".
The Afghanistan Human Rights Commission also expressed outrage, with executive director Mohammad Musa Mahmodi saying: "We condemn any killings done without proper trial. It is un-Islamic and against any human rights values."
It comes as US Secretary of State Hillary Clinton made a plea for the rights of women in Afghanistan, using a global forum to insist that they must be part of the country's future growth.
"The United States believes strongly that no nation can achieve peace, stability and economic growth if half the population is not empowered," she said.
Ms Clinton said the way forward "must include fighting corruption, improving governance, strengthening the rule of law (and providing) access to economic opportunity for all Afghans, especially for women".
"All citizens need to have the chance to benefit from and contribute to Afghanistan's progress. The United States will continue to stand strongly by the women of Afghanistan."
She was addressing a world conference in Tokyo aiming to chart a way forward for the nation after Nato combat troops withdraw in 2014.
The talks have already raised pledges of $16bn (£10bn) in civilian aid for the conflict-torn nation over the next four years.