Overwhelmed emergency services warned residents to head for high ground or climb onto rooftops -- not into attics -- so they could be seen by rescue helicopters.
"It is bad and growing worse," said Texas Governor Greg Abbott, who added that the storm -- a huge hurricane that first crashed ashore on the Gulf Coast late Friday -- had inflicted billions of dollars in damage.
US President Donald Trump, who was spending the weekend at the Camp David presidential retreat, said he would visit the Lone Star State as soon as he could do so "without causing disruption."
"The focus must be on life and safety," he said in a series of tweets about the disaster, his first major domestic challenge since taking office in January.
At least three people have died since Harvey made landfall as a Category 4 hurricane, spawning tornadoes and lashing east and central Texas with torrential rains.
In Houston, a woman drowned when she left a car which had stalled in high water, city mayor Sylvester Turner said. Another man was found dead in a flooded Wal-Mart parking lot in Galveston, officials confirmed.
Local officials said Saturday that one person was killed when a house caught fire in the Rockport area, where Harvey made landfall with sustained winds of 130 miles (215 kilometers) per hour.
"This event is unprecedented & all impacts are unknown & beyond anything experienced. Follow orders from officials to ensure safety," the National Weather Service said on Twitter.
Houston began opening community centers to shelter people forced out of their homes, but the mayor appealed to residents to stay put and not call the 911 emergency line unless they faced a life-threatening situation.
"Do not get on the road. Even if there's a lull today, don't assume the storm is over," Turner said at a news conference. "The best way to keep from being stranded is to stay off the streets."
Rain measured in feet
The National Weather Service said about two feet (60 centimeters) of rain fell in Houston and nearby Galveston in a 24-hour period.
"If the highest floor of your home becomes dangerous ... get on the roof!" Houston's emergency management agency warned in a bulletin.
Flooding was expected to worsen as the most powerful storm to hit the United States mainland since 2005 lingers over the area.
"This appears to be either the worst or one of the worst floods Houston has ever had. We are measuring it not in inches but in feet," Abbott told CBS's "Face the Nation" show.
Harvey slowly weakened as it advanced, but it had the power to rip off roofs, flip mobile homes and leave hundreds of thousands of people in the dark on the Gulf Coast, home to some of the country's most important oil refineries.
Tornado warnings were in effect in several parts of the area.
Hobby International, one of Houston's two airports, announced that all flights were canceled "due to standing water on runways," while George Bush International was operating at limited capacity.
The city's Ben Taub hospital was being evacuated after basement flooding caused power problems, local officials said.
Abbott said National Guard troops were deployed overnight in Houston, using high-clearance vehicles to help with rescues in inundated areas of the city of 2.3 million. Boats and helicopters also were being deployed.
The Coast Guard has so far rescued at least 100 people from the air.
Search and rescue operations were also underway in devastated coastal communities including Rockport, Aransas Pass, Port Aransas and Corpus Christi, a city of some 325,000 people, Abbott said.
In Victoria, a town just north of Rockport, residents were shocked by the storm's intensity.
"If I knew it was going to be what it came to be, I might have left sooner," local resident Robby Villa told AFP.
'Landmark' disaster
Abbott visited a shelter for coastal evacuees late Saturday in the state capital Austin and handed out food, describing the damage to homes and property as "sheer tragedy."
"Some of them had their homes mowed down. Some of them will not have a place to return to... It is our job to make sure they will be taken care of," he said.
The Federal Emergency Management Agency (FEMA) said there should be no illusions about the long-term impact.
"This disaster will be a landmark event," FEMA director Brock Long said on CNN's "State of the Union," adding that the agency is gearing up for a two-year recovery effort.
Oil industry prepared
Coastal Texas is a fast-growing area, with some 1.5 million people moving into the region since 1999. It is also home to a large number of oil refineries and a number of major ports.
US authorities said about 22 percent of crude production in the Gulf of Mexico, accounting for more than 375,000 barrels a day, was shut down as of Friday.
But Abbott said the oil industry was well prepared.
"They hunkered down and were able to contain the facilities, and they have the ability to ratchet up back up there quickly," he said on Fox News Sunday, predicting a "one- or two-week downturn."
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