"Emergency teams are mobilizing to offer all their support to those who have been affected," said Ecuadorian President Guillermo Lasso in a tweet.
The quake, which the U.S. Geological Survey (USGS) measured at magnitude 6.8, struck at a depth of 66.4 km (41.3 miles) about 10 kilometers (6.2 miles) from the city of Balao in the province of Guayas.
The earthquake did not appear likely to generate a tsunami, authorities said.
The presidency's communication agency said that the quake left 11 people dead in the province of El Oro and one fatality in Azuay province, while multiple people were being treated for injuries in hospitals.
The agency also said multiple homes, educational buildings and health centers had been damaged and that multiple roadways were blocked by landslides caused by the earthquake. The Santa Rosa airport suffered minor damage, but remained in operation.
Ecuador's Secretariat of Risk Management said in an earlier statement that the death in Azuay province occurred when a wall collapsed on to a vehicle. In other provinces, structural damage included a collapsed wharf and a collapsed wall in a supermarket.
The agency said that state-run oil company Petroecuador had evacuated and suspended activities in multiple facilities out of precaution, but had not reported damage.
"We all ran out into the streets... we were very scared," Ernesto Alvarado, a resident of Isla Puna near the epicenter, told Reuters, adding that some homes had collapsed.
The initial quake was followed by two weaker aftershocks in the following hour, according to the Geophysics Institute of Ecuador.
Peruvian authorities said that the quake was felt in the country's northern region, and that there were no immediate reports of harm to people or structures.
(Reporting by Alexandra Valencia in Quito, Fabián Andrés Cambero in Santiago, Jackie Botts in Mexico City; editing by Diane Craft, Josie Kao and Chizu Nomiyama)
By Alexandra Valencia
