The first typhoon to directly hit Taiwan in four years, Haikui made landfall on Sunday in the island's mountainous and sparsely populated far southeast, before moving across the south.
It knocked out power to more than 240,000 households, state-run utility Taipower said, but by Monday, fewer than 34,000 were still waiting for electricity to be restored, about half of them in the eastern county of Taitung.
Classes were canceled and workers given the day off across southern, eastern and central Taiwan, while Taipei, the capital, received sporadic gusty rain showers.
The world's largest contract chipmaker, Taiwan Semiconductor Manufacturing (TSMC), said its plants in Taiwan were operating normally and had not been affected by the storm.
Fire officials reported five injuries from the typhoon but were still trying to ascertain if the death of a man found by a roadside in Taitung was linked to it.
Taiwan airlines canceled 208 domestic flights, leaving just a handful scheduled, while ferry services to surrounding islands were suspended.
International flights, with just 23 canceled, suffered less disruption, the Civil Aeronautics Administration said.
Haikui is much weaker than Typhoon Saola, which hit Hong Kong and the southern Chinese province of Guangdong on Saturday.
By Monday, Haikui had started to enter the Taiwan Strait, heading for mainland China, Taiwan's weather authorities said, though it will continue to bring heavy rain across the island into the middle of the week.
China's national weather and ocean forecasters have warned of strong winds and large waves around the coastal provinces of Fujian and Guangdong and urged ships to take precautions, Chinese state media said.