One person also died in neighbouring Guatemala, its president has said.
The quake, measured at 8.1 by the US Geological Survey but higher by Mexico, struck in the Pacific, about 87km (54 miles) south-west of Pijijiapan.
A tsunami warning has been issued for Mexico, Guatemala, El Salvador, Costa Rica, Nicaragua, Panama and Honduras.
The quake was felt in Mexico City, with buildings swaying and people running into the street. The tremors there, about 1,000km from the epicentre, were reported to have lasted up to a minute.
Mexican President Enrique Peña Nieto said some 50 million Mexicans would have felt the tremor and that the death toll might rise.
Three deaths were reported in Mexico's Chiapas state, near the epicentre, and two more in Tabasco state.
Severe damage was reported to buildings in southern Mexico and in Guatemala.
The Pacific Tsunami Warning Center said that tsunami waves "reaching more than three metres above the tide level are possible along the coasts of Mexico". There is a coastal evacuation in Chiapas state.
The quake struck at a depth of 70km, the USGS said.
At magnitude 8.1, the quake outstrips the deadly 1985 tremor that hit close to Mexico City and caused thousands of deaths.
The Mexican interior ministry has given the latest quake a higher magnitude, of 8.4.
More than 10 aftershocks ranging from 4.3 to 5.7 in magnitude have been recorded closer to the Mexican coast, off the town of Paredon.
Mr Peña Nieto warned there might be more.
He also said the Salina Cruz refinery on the southern coast had temporarily suspended operations.
Schools have been closed in 11 Mexican states.
Some electricity cuts have been reported in the capital and social media video showed lampposts swaying violently, but there are no reports of major damage there.
Journalist Franc Contreras, who is in Mexico City, told the BBC: "You could hear loud cracks in the concrete. It sounded like a giant wooden branch being just broken open violently.
"People were streaming out of the hallways here. And everybody walking out single file into the streets, trying to avoid overhead high power lines."
Guatemalan President Jimmy Morales appealed for calm on national television and in a Twitter post.
"We have reports of some damage and the death of one person, even though we still don't have exact details," Associated Press quoted Mr Morales as saying.
No tsunami warning has been issued for the US west coast.
Mexico is currently also being threatened on its eastern coast by Hurricane Katia.
The category one hurricane is about 300km south-east of Tampico and has sustained winds of 140km/h the National Hurricane Center says.