Officials say a dust cloud caused the pilot to lose control, keel over and crash into an open field where local residents had set up camp for the night because they feared aftershocks could cause their homes to collapse.
Twelve people died at the scene of the crash in the southern state of Oaxaca and another person later died in hospital, the state prosecution service said.
Mexico's Interior Department said the officials were evaluating reports of damage from the quake.
Interior Minister Navarete told TV network Televisa that he and Oaxaca Governor Alejandro Murat were not seriously harmed in the accident that happened as the helicopter was attempting to land.
The pilot lost control shortly before landing, the Interior Ministry said.
The Defense Department said the Blackhawk helicopter crashed as it was preparing to land in a vacant lot. The victims had apparently been waiting below to greet the minister. "In that moment, you couldn't see anything, nothing else was heard beside the sound that iron makes when it scrapes the earth".
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Navarrete and the defense department said they regretted the loss of life.
Two people suffered fractures and non-life threatening injuries in Pinotepa Nacional.
A powerful natural disaster shook Mexico City on Friday, triggering the capital's quake alarm system and causing buildings to sway, reported AFP.
Navarrete has led the Interior Department, which oversees most political and security issues, for little over a month.
The quake was powerful but struck at a relatively deep 24.7 km, according to the U.S. Geological Survey. It said the epicenter was 33 miles northeast of Pinotepa in southern Oaxaca state.