Haibatullah Akhunzada, a religious scholar who was named in a United Nations report last year as the Taliban's former chief justice, will lead the movement, it said in a statement.
Sirajuddin Haqqani, head of a feared network blamed for many deadly bomb attacks in Kabul in recent years, and Mullah Mohammad Yaqoob, son of Taliban founder Mullah Mohammad Omar, will serve as deputies.
The announcement, following a meeting of the Taliban's main shura, or leadership council, ended days of confusion during which the Taliban declined to confirm the death of Mansour in a drone strike in Pakistan on Saturday.
“All the shura members have pledged allegiance to Sheikh Haibatullah in a safe place in Afghanistan,” the statement said. "All people are required to obey the new Emir-al-Momineen (commander of the faithful).”
Akhunzada, believed to be around 60 years of age and a member of the powerful Noorzai tribe, was a close aide to Omar and is from Kandahar, in the south of Afghanistan and the heartland of the Taliban.
An official Taliban account on Twitter posted a photograph of Akhundzada, informally known as Mullah Haibatullah, with a white turban and a long, greying beard.
The post listed his full title as Amir-ul-Mumineen Shiekh ul Quran Haibatullah Akhundzada.
The hardline Taliban movement banned human images for breaching their strict interpretation of Islam during their five-year rule over Afghanistan, which ended when they were ousted by a US-led military campaign.
Questions over peace talks
Senior members of the insurgent group had been keenly aware of the need to appoint a candidate who could bring disparate factions together and repair the splits that emerged last year when Mansour was appointed.
However, there was no immediate indication of whether the appointment would lead to a shift in the stance of the Taliban, which under Mansour ruled out participating in peace talks with the government in Kabul.
A spokesman for Afghan Chief Executive Abdullah Abdullah called on the new Taliban leader to join talks, or face dire consequences.
“We invite Mula #Haibatullah to peace. Political settlement is the only option for #Taliban or new leadership will face the fate of #Mansoor,” spokesman Javid Faisal said in a tweet.
We invite Mula #Haibatullah to peace. Political settlement is the only option for #Taliban or new leadership will face the fate of #Mansoor .— Javid Faisal (@Javidfaisal) May 25, 2016
The United States, Pakistan and China have also been trying to get the militants to the negotiating table to end a conflict that has killed thousands of civilians and security personnel and left Afghanistan seriously unstable.
News of the appointment came as a suicide attack on a bus carrying staff from an appeal court killed 10 people and wounded four west of the Afghan capital, Kabul. There was no immediate claim of responsibility.
The Taliban have made big gains since NATO forces ended their main combat operations in Afghanistan in 2014, and now control more of the country than at any time since they were ousted by US-led forces in 2001.
Mansour, a former deputy to Omar named as leader after the Taliban announced that Omar had died more than two years earlier, faced widespread suspicions that he had deceived the movement by covering up his predecessor's death.