All the deaths were in three neighbouring states clustered near the epicenter that lay about 70km off the coast.
The states of Chiapas, and Oaxaca in southern Mexico were closest to the quake.
The worst-hit city appeared to be Juchitan, on the narrow waist of Oaxaca known as the Isthmus.
Emergency crews in Mexico have recovered the body of a police officer buried in rubble, raising to 65 the death toll from this week's 8.1-magnitude natural disaster.
The quake, which hit minutes before midnight on Thursday, toppled hundreds of buildings, triggered tsunami evacuations and sent panicked people fleeing into the streets in the middle of the night. That includes 37 in Juchitan.
Mexico's President Enrique Pena Nieto has confirmed that the death toll from Thursday night's 8.1 magnitude quake on the southern coast has now reached 65.
"It's 71 (dead). Just for Oaxaca", said Jesus Gonzalez, a spokesman for the state civil protection authority.
While in Chiapas, the Mexican Interior Department reported that 1,700 houses have been damaged and over 400 are completely destroyed.
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Katia was lashing the state of Veracruz, which borders the Gulf of Mexico, as well as parts of Hidalgo and Puebla.
At least one person died when a church collapsed in San Sebastian, Guatemala, according to The Independent. The mountainous region where the storm has been dumping rain has a history of deadly floods and slides.
State authorities' main priorities are searching people trapped in the wreckage and securing shelter for those forced from their homes, Murat said.
Police, soldiers and emergency workers raced to rescue survivors from the ruins of Mexico's most powerful natural disaster in a century, which killed at least 61 people, as storm Katia menaced the country's eastern coast yesterday with heavy rains. A state senator from Chiapas told The Daily Beast, "We could be looking at more than 10,000 people facing crisis".
It weakened as it made landfall on Friday night. The quake shook a large swath of the country and was felt as far north as Mexico City.
The U.S. National Hurricane Center says that Tropical Storm Katia is starting to stall over Mexico's Sierra Madre mountains.
The US National Hurricane Centre said Hurricane Katia's maximum sustained winds had dropped on Saturday morning to about 56km/h.
The hurricane center said that Katia's maximum sustained winds are now down to near 40 miles per hour (65 kph) and that it is expected to drift east of the Sierra Madre mountains as it peters out.