The quake on Sunday initially sparked a tsunami warning for some coasts in Peru and neighbouring Chile, but this was quickly called off.
Its epicentre was in the Lomas district, in the southern region of Arequipa.
Arequipa governor Yamila Osorio said on Twitter that a 55-year-old man had been killed after being struck by a falling rock, while 20 others suffered "light injuries" in the town of Cacha.
Jorge Chavez, chief of Peru's Civil Defense Institute, told local radio station RPP that a second death was reported in the town of Bella Union.
Several municipalities have been left without electricity, while many roads and houses have collapsed, Ms Osorio said.
Residents of Lomas were evacuated after feeling an aftershock, she said.
Shortly after the earthquake, which struck shortly before 4.20am local time (9.20am in the UK), the Pacific Tsunami Warning Center warned of "hazardous tsunami waves".
But a later message said "there is no longer a tsunami threat from this earthquake" and that the centre had not observed any tsunami waves.