Jackson Sanford Staing confessed Sunday night that he murdered Leslie Dayana Paredes Silva at the prison.
The woman's body was located and taken to the morgue in Lima, National Penitentiary Institute chief Wilson Hernandez told Radio Programas del Peru, or RPP.
The inmate argued with the woman on visiting day and strangled her, Hernandez said.
The Dutchman buried Paredes Silva in his cell and later constructed a cement bench over the grave.
Prison personnel constantly monitor cells to ensure that unauthorised people do not remain in them after visits, but no one noticed that the woman did not leave the facility back in August, Hernandez said.
The Lurigancho prison, located in Lima's San Juan de Lurigancho district, is one of the largest in Peru, holding more than 8,000 inmates.
The penitentiary also holds some of the most dangerous inmates in the Andean country.
Prosecutors and the National Police, which is in charge of security at the prison, are investigating the murder, Hernandez said.