Kong-Rey skirted the island's east coast before heading north towards Japan. It is expected to fizzle out over the western Pacific without causing additional damage.
Particularly hard hit in Taiwan were the large west-coast cities of Chiayi, Tainan and Kaohsiung, where flooding in some areas reached second-storey levels.
In Tainan, officials evacuated 29 people from a nursing home and cancelled some train services, while cars were stranded in nearby Chiayi.
There were no immediate reports of casualties from the storm.
A worker at the nursing home said: "The water quickly came up as deep as waist height and it was flooded in about 15 minutes."
High winds caused scores of international flights to be cancelled and, in conjunction with the rain, led to the disruption of the high-speed rail service between the capital Taipei and Kaohsiung.
Most offices and schools were closed in southern Taiwan and the military stepped up its preparations, ordering more than 50,000 troops to be on standby.
Kong-Rey is the second major storm to hit Taiwan this month.
Last week, severe tropical storm Trami dumped up to 39 ins of rain on the southern part of the island.