Pictures show landslides and ruined mud-built homes in eastern Paktika province, where rescuers have been scrambling to treat the injured.
Taliban leader Hibatullah Akhundzada said hundreds of houses were destroyed and the death toll was likely to rise.
It is the deadliest earthquake to strike Afghanistan in two decades.
The head of information for Paktika province, Mohammad Amin Hazifi, told the BBC that 1,000 people had died and 1,500 are injured.
Rescue teams are still searching for others buried under the ground, he added
The earthquake struck about 44km (27 miles) from the south-eastern city of Khost shortly after 01:30 local time (21:00 Tuesday GMT), when many people were asleep at home.
“Every street you go, you hear people mourning the deaths of their beloved ones. Houses are ruined,” a local journalist in badly-hit Paktika province said to the BBC.
Earthquakes tend to cause significant damage in Afghanistan, where dwellings in many rural areas are unstable or poorly built.
Local farmer Alem Wafa cried as he told the BBC that official rescue teams are yet to reach the remote village of Gyan – one of the worst hit.
“There are no official aid workers, but people from neighbouring cities and villages came here to rescue people. I arrived this morning, and I – myself – found 40 dead bodies.”
He continued: “Most of them are young, very young children. There is a hospital here. But it can’t deal with this disaster. It doesn’t have the capacity.”
Taliban officials called for aid agencies to rush to the affected areas in the nation’s east.
Decades of conflict have made it difficult for the impoverished country to improve its protection against earthquakes and other natural disasters – despite efforts by aid agencies to reinforce some buildings over the years.
Even before the Taliban takeover, Afghanistan’s emergency services were stretched to deal with natural disasters – with few aircraft and helicopters available to rescuers.
Communication following the quake is difficult because of damage to mobile phone towers and the death toll could rise further still, another local journalist in the area told the BBC.
“Many people are not aware of the well-being of their relatives because their phones are not working,” he said. “My brother and his family died, and I just learned it after many hours. Many villages have been destroyed.”
In remote areas, helicopters have been ferrying victims to hospitals.
Speaking to the BBC, a doctor from one of the worst-hit districts in Paktika province said medical workers were among the victims.
“We didn’t have enough people and facilities before the earthquake, and now the earthquake has ruined the little we had,” they said. “I don’t know how many of our colleagues are still alive.”
Speaking to Reuters news agency, locals described horrific scenes of death and destruction in the aftermath of the late-night earthquake.
“The kids and I screamed. One of our rooms was destroyed. Our neighbours screamed and we saw everyone’s rooms,” said Fatima.
“It destroyed the houses of our neighbours,” local resident Faisal said. “When we arrived there were many dead and wounded. They sent us to the hospital. I also saw many dead bodies.”
Most of the casualties so far have been in the Gayan and Barmal districts in Paktika, a local doctor told the BBC. Local media site Etilaat-e Roz reported a whole village in Gayan had been destroyed.
The earthquake was magnitude 6.1 at a depth of some 51km, according to seismologists.
Tremors were felt across more than 500km of Afghanistan, Pakistan and India. Witnesses reported feeling the quake in Afghanistan’s capital, Kabul, as well as Pakistan’s capital, Islamabad.
However, there have been no immediate reports of casualties or significant damage in Pakistan, according to BBC Urdu.
Afghanistan is prone to quakes, as it’s located in a tectonically active region, over a number of fault lines including the Chaman fault, the Hari Rud fault, the Central Badakhshan fault and the Darvaz fault.
Over the past decade more than 7,000 people have been killed in earthquakes in the country, the UN’s Office for the Co-ordination of Humanitarian Affairs reports. There are an average of 560 deaths a year from earthquakes.