The weather bureau said the epicentre on Sunday was in Taitung county, and followed a 6.4 magnitude tremor on Saturday evening in the same area, which caused no casualties.
The US Geological Survey measured Sunday's quake at a magnitude 7.2 and at a depth of 10 kilometres.
Taiwan's fire department said two people were trapped in a building housing a convenience store that collapsed in Yuli and two people had been rescued, while three people whose vehicle fell off a damaged bridge were rescued and taken to hospital.
The Taiwan Railways Administration said three carriages came off the rails at Dongli station in eastern Taiwan after part of the platform canopy collapsed. The fire department said one person was injured.
About 70 people are also trapped on the scenic Chike mountain area by a blocked road, though there are no injuries, the department added.
The US Pacific Tsunami Warning Center issued a warning for Taiwan after the tremor but later lifted the alert. Japan's weather agency lifted a tsunami warning for part of Okinawa prefecture.
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 Semiconductor Manufacturing Co (TSMC) , the world's largest contract chipmaker, said there was "no known significant impact for now".
Taiwan lies near the junction of two tectonic plates and is prone to earthquakes.
More than 100 people were killed in southern Taiwan in 2016, while a 7.3 magnitude earth quake killed more than 2,000 people in 1999.
