The men who escaped from Pul-i-Charki prison on the outskirts of Kabul on Wednesday were facing sentences ranging from death to 15 years in jail after being convicted of terror offences, officials said on condition of anonymity.
They did not say if the three were linked to the al Qaeda or Taliban groups active in Afghanistan or give details of the charges.
The interior ministry refused to comment, saying it was trying to collect details.
The escape echoed one in January in which seven low- to mid-ranking Taliban militants fled the jail by pretending to be visitors. Inside help was alleged in that incident.
Officials identified the three men involved in the latest escape as: Abdul Bari who had been sentenced to death; Abdul Qadir given a 20-year term; and a man identified only as Naqibullah, sentenced to 15 years.
"The trio escaped the prison yesterday at visiting hours pretending to be visitors," an interior ministry official said.
Another police source confirmed the escape and said one prisoner had also been found dead in mysterious circumstances that may have been linked to the escape.
Pul-i-Charkhi holds criminals as well as captured low-ranking Taliban and al Qaeda members, including foreign fighters.
Most of the militants rounded up after the US-led invasion that toppled the Taliban regime in 2001 were transferred to the United States' Guantanamo Bay jail or one at a base in Bagram, north of Kabul.
A four-day revolt at the prison in February left around five prisoners dead. Officials said the revolt was instigated by about 300 Taliban and al Qaeda prisoners trying to create chaos so they could escape.
The massive and rundown jail is notorious for the detention and torture of thousands of people during the communist rule of the 1980s.

Copyright AFP (Agence France-Presse), 2006