The regional head of police, General Walter Ortiz, said that in an interview that 43 had died at the site of the crash, after the bus tumbled down a 200-meter ravine, while one person died in a nearby hospital.
Peruvian radio, RPP, updated the death count from 30 to 35 then to 44, while another 34 people were injured.
The severely injured were airlifted by helicopter to the city of Arequipa by the Peruvian army.
The accident occurred around 1:30 a.m. at kilometer 782 of the South Panamerican Highway.
The bus had left on Tuesday night from the city of Chala to reach Arequipa, with 45 passengers on board, but an unknown number of passengers joined during the journey.
The regional head of civil defense for Arequipa, Jacqueline Choque, said that there was no exact headcount for the number of people on board at the time of the crash.
Initial inquiries found that the driver seems to have fallen asleep and lost control. This is the second major bus crash in Peru this year, after another accident killed 52 people north of Lima.