The two injured people were traveling in a van when a tree fell on the vehicle, “leaving them trapped inside,” CNA said.
They suffered injuries but were conscious and taken to hospital, it added.
The storm made landfall along the southeastern coastal township of Donghe at about 3:40 p.m. local time, the weather bureau said.
As of 2 p.m. local time on Sunday, more than 9,300 households in the Taitung and Hualien counties had lost electricity, state electricity provider Taipower said. Repair works are underway.
The capital Taipei is currently experiencing heavy rains.
The weather bureau said Typhoon Haikui was moving westwards at a pace of 7 kilometers (4 miles) per hour with winds of up to 155 kilometers per hour (96 miles per hour).
Haikui is the first typhoon in four years to directly hit Taiwan and is expected to bring heavy rainfall to eastern regions, according to the weather bureau.
It warned that heavy rainfall could trigger flash flooding and landslides in parts of the country. The mountainous Hualien county could see total precipitation of more than 500 millimeters, it added.
The typhoon was previously expected to make landfall near the southeastern city of Taitung around 5 p.m. local time on Sunday.
As of 7 a.m. local time Sunday, more than 2,800 residents within seven counties and municipalities had been evacuated, the island’s Ministry of the Interior said.
Seven counties and cities in southern and eastern Taiwan have suspended school and work for Sunday, according to the respective local governments. In addition, Yunlin county and the outlying Penghu county will suspend school and work from Sunday noon.
The typhoon has led to the cancellation of 222 domestic flights and 37 international flights departing from Taiwan, according to Taiwan’s Ministry of Transportation and Communications.
In neighboring Hong Kong, at least 86 injuries were reported from another typhoon, Saola, which made landfall in the city after passing through the northeastern Philippines.
In an update on Saturday, Hong Kong authorities said they had received more than 1,500 reports of fallen trees, 21 cases of flooding and two reports of landslides.
That typhoon has since weakened to a tropical storm but left in its wake hundreds of canceled and disrupted flights.
With previous reporting from CNN’s Chris Lau, Tara Subramaniam, Nectar Gan and Jessie Yeung in Hong Kong
