He said people were evacuating homes in Lomas district, where the epicentre was located, 42 kilometres south-southwest of Acari.
Sunday's quake is a glimpse at how bad a bigger natural disaster in a more populated part of Peru would be.
INDECI also said that 65 had been injured in the quake.
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Peru's government Geophysical Institute said the quake was of magnitude 6.7 with its epicentre in Lomas, in the southern region of Arequipa. Acari is about an eight-hour drive down the coast from Lima, the capital.
The quake destroyed adobe structures and killing at least one person, according to officials.
The Lomas de Lachay (Lachay Hills) is a national reserve in the desert foothills of Huaura Province in Lima region of Peru. The last time Peru was struck by a massive natural disaster was in November 2015, when a 7.6 quake struck northwest of Iberia.
Peru is the world's number two copper producer, although numerous mines in the south are located far inland from where the quake struck. The center later issued a bulletin saying no such waves were detected and the threat had passed.
The Pacific Tsunami Warning Center says that "hazardous tsunami waves are forecast for some coasts". Chile's navy said the quake did not meet conditions to produce a tsunami, Reuters reports.
Earthquakes are common along Peru's coast.
The natural disaster, which struck just after 4 a.m. local time, was centered near the coast of Peru, about 25 miles (40 kilometers) south-southwest of Acari, according to the US Geological Survey.
Ms Osorio said on Twitter the quake caused minor material damage, including some collapsed roads.