Typhoon Soulik has brought strong winds and torrential rain to the island.
So far one person is reported to have died while 21 have been injured in the extreme weather.
More than 8,500 people were evacuated from mountainous and other dangerous areas and thousands of soldiers were deployed.
Some 300,000 people were evacuated in China as the medium-force typhoon approached the eastern provinces of Fujian and Zhejiang.
Local authorities there have been asked to implement emergency response plans, China's state-run news agency Xinhua reported, after recent torrential rain across large parts of the country reportedly left 200 people dead or missing.
Typhoon Soulik had wind speeds of around 173 km/h (100 mph) on Saturday morning.
It made landfall in Taiwan at around 03:00 on Saturday (19:00 GMT on Friday), Taiwan's Central Weather Bureau reported.
A police officer was killed by falling bricks but other people suffered mostly light injuries, including from fallen trees or being blown off their scooters.
The strong winds and heavy rain have caused electricity disruptions, a run on food and essential supplies in supermarkets, and uprooted trees and signs in some areas.
This typhoon was the first to hit Taiwan this year and there had been fears of major damage because the island was the first place it made landfall, reports the BBC's Cindy Sui in the capital Taipei.
Nearly 50,000 soldiers have been put on standby, she says.
Schools and offices in Taipei and several other cities had closed on Friday afternoon as the tropical storm neared.
Some flights to Taiwan were disrupted, with both Cathay Pacific and China Airlines announcing cancellations.
Precautionary measures were taken to close the roads and bridges along areas most susceptible to disaster, officials said.
Fishing boats had been returned to the shore before the typhoon hit, and members of the public were urged to avoid mountain and coastal areas.
Evacuated residents - including 3,000 from Kaohsiung city and 2,000 from Pingtung county in the south of Taiwan - were taken to local government buildings that have been turned into shelters, Agence France Presse reported.
More than 2,000 tourists had earlier been evacuated from Taiwan's Green Island, near the city of Taitung, as a precaution.
Typhoons are common during the summer in parts of East Asia, where the warm moist air and low pressure conditions enable tropical cyclones to form.
In 2009, Taiwan was hit by Typhoon Morakot, which left hundreds dead in floods and mudslides.