Dujuan hit the island's northeastern tip earlier than expected after picking up speed as it approached, with more than 7,000 people evacuated in advance.
Taiwanese authorities said tens of thousands of homes were without power in the north.
Waves crashed along the east coast and the capital Taipei was swept with wind and rain, but there were no immediate reports of injury.
"It picked up speed in the past few hours, but will slow somewhat after hitting the mountains in the east," an official from the Central Weather Bureau said.
Dujuan made landfall in the northern county of Yilan, where some areas have seen more than 500 millimetres (20 inches) of rain in the last 24 hours.
Panicked visitors to the island's east -- many of whom had headed there for the Mid-Autumn Festival long weekend -- crammed onto trains away from the eastern cities of Hualien and Taitung before services were suspended.
High-speed rail services in the west were also cancelled.
Authorities warned that coastal areas could be particularly dangerous as tides are affected by the current "supermoon" -- a rare astrological event in which the moon appears brighter and larger.
This is because the moon has reached its closest orbital point to Earth and therefore has a stronger gravitational pull than usual.
The storm intensified as it approached Taiwan, with gusts of 227 kilometres (141 miles) per hour.
Damage threat
Dujuan was graded a "strong typhoon" -- the top category -- by Taiwan's weather bureau while other regional forecasters, including the Hong Kong Observatory, categorised it as a "super typhoon".
China issued a red alert, its highest, for Dujuan.
Heavy rain and gales are set to hit parts of Fujian, Zhejiang and Shanghai, the official Xinhua news agency quoted the National Marine Environmental Forecasting Center as saying.
The centre instructed ships' crews to return to harbour and residents to stay indoors.
Almost 3,000 people were evacuated Sunday from Taiwan's Green Island and Orchid Island -- popular with visitors.
More than 4,000 were moved Monday ahead of the storm.
New Taipei City Mayor Eric Chu said they were from vulnerable areas, including the hot springs town of Wulai just outside Taipei.
Aboriginal mountain communities are particularly at risk during typhoons, often affected by flooding and mudslides.
Wulai was hit hard by Typhoon Soudelor in August, with some residents unable to return home for weeks.
Shops and hotels have remained closed as roads and flood damage are yet to be fully repaired.
"To be honest, we all feel very depressed. Any damage may further prolong the time needed for reconstruction," Chou Chih-kang, a Wulai neighbourhood chief, told AFP.
By Monday evening Wulai had already seen 450 mm of rain in 24 hours.
The weather bureau warned that the "massive amount or rubble" on mountain slopes and riverbeds since Soudelor could lead to further damage.
More than 24,000 troops are on standby for disaster relief and evacuations, with 100 shelters set up. Emergency response centres have been established in the north and east.
A concert by US rock band Bon Jovi due to take place in Taipei Monday was cancelled, while 169 international and 59 domestic flights were also pulled.
Ferry services and flights to outlying islands were suspended.
Dujuan passed near the Japanese island of Ishigaki as it approached Taiwan.
Around 100 domestic flights were cancelled in Japan, while 3,200 households lost power in Ishigaki and other islands, local media said.
Typhoon Soudelor caused at least eight deaths in Taiwan last month and killed 21 people in China.
source: www.abs-cbnnews.com
