Two people have so far been killed, including an infant girl, local TV reports said.
Rescuers mounted hydraulic ladders and a crane to scour the wreckage and pluck more than 120 survivors to safety, with at least 26 taken to hospital, a fire brigade official said.
One elderly woman, wrapped in blankets, was strapped to a board and slowly slid down a ramp to the ground as the cries of those still trapped rang out. Rescuers used dogs and acoustic equipment to pick up weaker signs of life among the rubble.
"There are 60 households in that building," said Tainan City Government Fire Bureau information officer Lee Po Min, estimating that there might be about 240 people living there.
The U.S. Geological Survey said the magnitude 6.4 quake was centered 27 miles (43 km) southeast of Tainan. It struck at a depth of 23 km (14 miles), revised from an initial estimate of 10 km. Shallow quakes can be more devastating than deeper tremors. At least five aftershocks of 3.8-magnitude or more shook Tainan about half an hour after the initial quake, according to Taiwan's Central Weather Bureau.
Several other buildings had also been damaged, including one that was pictured leaning at an alarming angle over a street in Tainan, a city of 2 million people, many of whom lived through a massive 1999 tremor that killed about 2,400 people.
Taiwan lies in the seismically active "Pacific Ring of Fire". Local TV quoted Tainan residents as saying the quake felt worse than the 1999 tremor, centered in central Taiwan.
Taiwan's Formosa TV said its reporters could hear the cries of some residents trapped inside the collapsed apartment tower as firefighters, police and troops swarmed the area.
Firefighters hosed down part of the building to prevent a fire while others used ladders and a crane to enter upper floors. The building appeared to have collapsed onto the first story where a child's clothes fluttered on a laundry line.
Some bullet train services were suspended to the south of Taiwan as inspections were carried out on the tracks for any quake damage, Taiwan High Speed Rail Corp said in a statement.
"Collapsed buildings reported in Tainan, with rescue workers arriving on scene. The city government there has set up a level one emergency response center. Onlookers are urged not to block access to emergency crews moving into the area," Taiwan's China Post newspaper said on its Web site.
Liu Shih-chung, an official with the Tainan City Government, said the city had set up an emergency response center.
According to the USGS, the last time Taiwan was struck by a quake of the same magnitude as Saturday's was in April 2015, but that temblor was deeper.