The weather bureau said the epicenter was in Taitung County and followed a 6.4 magnitude temblor on Saturday evening in the same area, which caused no casualties.
The US Geological Survey measured the quake at a magnitude 7.2 and at a depth of 10 kilometers (6.2 miles).
Taiwan media said a low-rise building housing a convenience store collapsed, and rescue work had begun to free those inside.
At least one train carriage was derailed at a station on the east coast. The railway authority said there were no casualties.
The US Tsunami Warning Center issued a warning in Taiwan after the tremor. It said hazardous tsunami waves were possible within 300 kilometers (190 miles) of the epicenter along the coasts of Taiwan.
Japan’s weather agency issued a warning for tsunami waves of 1 meter for part of Okinawa prefecture following the earthquake.
The quake could be felt across Taiwan, the weather bureau said. Buildings shook briefly in the capital Taipei.
Science parks in the southern cities of Tainan and Kaohsiung, home to major semiconductor factories, said there was no impact on operations.
Taiwan lies near the junction of two tectonic plates and is prone to earthquakes.
More than 100 people were killed in a quake in southern Taiwan in 2016, while a 7.3 magnitude quake killed more than 2,000 people in 1999.