Police say the bus was travelling on a dangerous stretch of mountain road when the driver lost control of the vehicle in the wet.
The injured were taken to two separate hospitals in nearby Arequipa, with most of them being treated at the Honorio Delgado de Arequipa regional hospital.
The bus was carrying 42 students to the World Social Forum in Venezuela. The students had left Belo Horizonte on January 9 and were travelling through Bolivia to Peru. They were due to arrive in Caracas on January 29.
Hundreds of people die in bus crashes every year in Peru because of perilous roads, poorly maintained vehicles and driver error in poor weather conditions.
At least 24 people have died in bus crashes so far in January, according to local transit police data.