William Lai, Tainan Mayor, speaking at the site of the disaster, hours after the earthquake struck the island early Saturday, said that the building complex housed 221 residents.
He said 31 other people were sent to hospital.
A girl and a man of 40, were confirmed dead, according to a government official. Local media reported a third death.
Rescuers mounted hydraulic ladders and a crane to search the debris and carried away more than 120 survivors to safety, with at least 26 taken to hospital, an fire brigade official said .
A local hospital said 58 people had been brought in, most of them with minor injuries.
Images posted on social media showed a collapsed building and several people trying to rescue those who were trapped. Another building was also shown in a partially collapsed state.
The US Geological Survey (USGS) said the earthquake was centred 43km southeast of the city of nearly two million people.
“I used a hammer to break the door of my home which was twisted and locked, and managed to climb out,” she told local channel SET TV, weeping as she spoke.
The shallow quake struck at a depth of 10 kilometres (six miles) around 2000 GMT Friday, according to the US Geological Survey, 39 kilometres northeast of Kaohsiung, the second-largest city on the island and an important port.
The quake was initially reported as having a magnitude of 6.7, but was downgraded to 6.4.
The Pacific Tsunami Warning Center said a destructive Pacific-wide tsunami was not expected.
Taiwan lies near the junction of two tectonic plates and is regularly hit by earthquakes.
A strong 6.3-magnitude quake that hit central Taiwan in June 2013 killed four people and caused widespread landslides.
A 7.6-magnitude quake struck the island in September 1999 and killed around 2,400 people.
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