Mehsud was killed when missiles struck his vehicle in the North Waziristan region on 1 November.
The government had been trying to set up peace talks, but the new leader has already rejected the initiative.
The BBC’s Richard Galpin in Islamabad says the Taliban have indicated that Mullah Fazlullah wants revenge for the killing of Mehsud.
A Taliban spokesman told the BBC the militants would target the military and the governing party.
The Mehsud killing had angered the Pakistani government. Interior Minister Chaudry Nisar Ali Khan said that the drone strike was “not just the killing of one person, it’s the death of all peace efforts”.
The announcement of the new leader was made by the Taliban’s caretaker leader Asmatullah Shaheen at a news conference at an undisclosed location.
When the news was announced, there was reportedly heavy celebratory gunfire in the area around Miranshah, the main town in the tribal area of North Waziristan.
Mullah Fazlullah led a brutal campaign in Swat between 2008 and 2009, enforcing hardline Islamic law, that included burning schools, and public floggings and beheadings.
A military operation was launched to retake the area.
Mullah Fazlullah fled over the border into Afghanistan but Islamabad says he has continued to orchestrate attacks in Pakistan.
He was accused of being behind a roadside bomb in September that killed Maj Gen Sanaullah Niazi, the top commander in Swat, along with two other military personnel.
Mullah Fazlullah was known for his radio broadcasts calling for strict Islamic laws and earning him the nickname “Mullah Radio”.
The shooting of Malala Yousafzai in October 2012 sparked outrage in Pakistan and across the globe.
The teenager had spoken out against the Taliban’s restrictions on girls’ education.
She was airlifted to the UK for hospital treatment and now lives in Birmingham with her family.
This year Malala, now 16, addressed the UN General Assembly and won the European Union’s Sakharov human rights prize.
Prior to the latest Taliban announcement, the BBC’s M Ilyas Khan in Islamabad said that Mullah Fazlullah was not a member of the Mehsud clan and, if appointed, would face a challenge to control the Mehsud fighters, who make up the bulk of the Taliban’s manpower.
The Taliban’s ruling council took seven days to reach a decision.
Prime Minister Nawaz Sharif had come to power in May pledging a negotiated settlement to the insurgency, but militant attacks continued.
The government said a delegation had been due to fly to North Waziristan to discuss peace talks with Hakimullah Mehsud but he was killed in the drone strike the day before.
There had been some hope a new leader of the Taliban would be more open to the peace initiative.
Regional Taliban commander Khan Said Sajna, said to favour such a move, had been touted as a favourite before the latest announcement.