After making landfall, the storm weakened with sustained winds of 155kph, the Central Weather Bureau said.
Sepat will continue to lose strength, but the storm could deluge western and southern Taiwan with torrential rains as it moves west towards China, the bureau said.
Authorities ordered the evacuation of thousands of people around the island as electricity supplies faltered and rains battered already saturated river valleys, mountainsides and built up areas.
South of Hualien city, the authorities ordered the evacuation of more than 1,000 people from the coastal community of Taidung, where electricity supplies were cut to 14,000 homes, a fraction of the 70,000 homes without power in eastern Taiwan.
Other evacuations, on a smaller scale, were ordered from mountainous areas in Taoyuan and Taipei counties in the north, and Nantou county in the centre of the island.
All domestic air and train services were halted and some international flights, including services to Hong Kong and Japan, was cancelled.
If Sepat stays on its current course, it will leave Taiwan late on Saturday and head for the heavily populated Chinese coastal province of Fujian.