The president described the quake as "the largest registered in our country in at least the past 100 years" - stronger even than a devastating 1985 natural disaster that killed more than 10,000 people in Mexico City.
The state government now says 15 people died there due to Thursday night's magnitude 8.1 quake, in addition to 45 reported dead in Oaxaca and four in Tabasco.
In Juchitan, emergency teams with search dogs found the body of a policeman under the rubble of city hall.
The Pacific Tsunami Warning Center said there is a possibility of widespread and unsafe waves on the coasts of Mexico, Guatemala, El Salvador, Costa Rica, Nicaragua, Panama, Honduras and Ecuador. But a preliminary analysis indicates the quake was triggered by the sudden breaking or bending of the Cocos plate, which dives beneath Mexico.
Officials said the death toll there could rise. In the end, the quake caused rough seas but no tsunami, officials said. This type of process doesn't happen often in subduction zones; usually, big quakes in subduction zones occur along the boundary between the sinking slab and the overriding crust.
The quake was felt as far away as Mexico City and Guatemala City.
The Category One storm, with sustained winds of 120 kilometers (75 miles) per hour, was closing in on the state of Veracruz which borders the Gulf of Mexico.
Israel will send aid to earthquake-hit Mexico in the coming days, coinciding with a previously planned visit to the North American country by Prime Minister Benjamin Netanyahu, the Foreign Ministry said Sunday.
Earlier, tsunami waves reaching 0.7 metres above the tide level were recorded by automated sea level gauges near the resort of Huatulco in Oaxaca state.
"The house moved like chewing gum and the light and internet went out momentarily", Rodrigo Soberanes, a resident of San Cristobal de las Casas in the Chiapas, told the AP.
Oaxaca Governor Alejandro Murat told Mexican television the quake hit 41 municipalities and had likely affected around one in five of the state's 4-million strong population.
On the outskirts of the city, the general hospital settled into its temporary home - a school gymnasium with gurneys parked on the basketball court.
Death toll rises to 33 in Mexico quake
Manuel Velasco said the 8.1 magnitude quake was the strongest on record in state history, topping a magnitude 7.9 quake in 1902. In Mexico City , people ran out of buildings after hearing quake warning sirens go off just before midnight (0500 GMT Friday).