The Federal Emergency Management Agency (FEMA) is reporting two people are injured but there are no reports of casualties.
The US National Weather Service said the slow-moving category 4 storm delivered rainfall of up to 5cm per hour overnight.
The weather service said that Mawar could produce landslides, flash flooding and life-threatening storm surges.
"Treat these imminent extreme winds as if a tornado was approaching and move immediately to an interior room or shelter NOW!," the weather service in Guam warned on Wednesday evening.
Even though the centre of the storm's eye passed just north of Guam, most of the island has lost power.
In an update early Thursday morning, the Service said the typhoon warning remained in effect for Guam and Rota Island in the Northern Marianas.
Nearly all of Guam Power Authority's circuits have been impacted by the storm and only about 1,000 of its 52,000 customers still had electricity, the authority said in a statement on Facebook.
"I'm very worried for our people's safety and very concerned," Guam's Governor Lou Leon Guerrero told media, while urging all residents to stay at home.
She compared the storm with the 1962 storm Typhoon Karen that flattened much of the US territory with a population of about 170,000, including about 10,000 US military personnel.
Houses damaged, people rescued
Guerrero said according to early reports, Mawar damaged houses and forced the rescue of eight people.
Images posted on social media showed ominous clouds drifting over beaches, rains lashing buildings and winds bending palm trees.
"Lots of us have relocated to the basement. All the units totally flooded, several windows blown out, and the building is shuddering from the wind," tweeted Ginger Cruz, who posted a video of winds overturning a pickup truck outside her home.
The northern third of Guam, including the village of Yigo, is being battered by the strongest winds and heaviest rainfall as the storm moves north-west, CNN reports.
President Joe Biden, who has approved an emergency declaration authorizing federal assistance for Guam, was briefed on the typhoon, White House spokesperson Karine Jean-Pierre said on Wednesday.
"The White House is in close contact with the government of Guam and has offered as much support as needed to this tragic, tragic major storm," Jean-Pierre said.
Meanwhile, the United States' Federal Emergency Management Agency (FEMA) has over 130 staff ready to help local local response efforts in Guam and the Northern Mariana Islands.
A FEMA spokesperson said over 100 generators as well as emergency communications equipment are being sent.
The spokesperson said FEMA's distribution centre in Guam is fully stocked with about four times the amount of food and water compared to the commodity stock in place during typhoon Mangkhut in 2018.
"The distribution centre is also connected to a network of warehouses and logistics capabilities, with another large warehouse in Hawaii," FEMA said.
"Additional commodities and supplies stand ready, if needed and requested."