"This is a life-threatening emergency," the city said on an emergency website. "Houston residents should avoid travel at all costs today."
Four of the deaths happened in hard-hit Harris County, officials said.
The driver of an 18-wheeler was found dead inside the cab after he drove into high waters, a Harris County constable reported, and another man was found dead in a submerged car, the Houston Fire Department said.
Two others were found dead after driving around a barricade on Houston's west side, Harris County Judge Ed Emmett said.
In neighboring Waller County, a 56-year-old man was found dead in a submerged vehicle, County Judge Trey Duhon said.
"It is believed the car rode off the road and into a ditch," Duhon said.
Crews performed 1,200 high-water rescues as of Monday afternoon, Harris County Emergency Management said.
"There's flooding in every part of Houston," Houston Mayor Sylvester Turner said, telling stranded citizens. "We will rescue you."
At least 1,000 homes were flooded in Harris County, according to Emmett. Some people were trapped in their homes or attics, fire department spokesman Ruy Lozano said.
The water was too deep for crews to reach them in high-water vehicles, so rescuers were trying to get to those people in boats, he said.
Nine hospitals in the region were closed to additional patients because of the flooding, the mayor said. Three apartment buildings were evacuated and residents were being sheltered at a mall, Turner added.
Emmett said Monday afternoon he estimates 240 billion gallons of rain has fallen in the Houston area. That estimate may soon rise: Rain and thunderstorms are forecast for Houston through late Tuesday, according to the National Weather Service.
A flash flood watch is in effect for the Houston area through Tuesday morning, with "life-threatening" flash flooding possible Monday night, the weather service said. As little as an inch of rain could aggravate the flooding, it said.
Emmett called it the most significant flood event since Tropical Storm Allison in 2001, which left 41 people dead. It caused more than $5 billion in property damage in Harris County alone, according to the county's Flood Control District.
At the Royal Phoenician apartment complex in north Houston, the brown floodwaters submerged cars in the parking lot early Monday, a resident said. He posted video to Twitter later showing the waters had gone down slightly, though levels still reached car windshields.
Another north Houston resident posted video of residents leaving their flooded homes in a canoe as the rain continued to fall.
Portions of I-10 and numerous roads throughout the metropolitan area were closed, as were many government offices. City bus and rail service shut down early Monday amid "severe and ever worsening weather conditions."
The storm snarled traffic at Houston's Hobby Airport, where nearly 200 flights were canceled by midafternoon, according to the airport's Twitter account.
Around 45,000 customers in the region were without power as of Monday afternoon, emergency management officials said, down from a peak of 123,000 earlier in the day.
Some areas had received as much as 16 inches of rain by Monday morning, according to the flood control district.
The heavy rains forced seven of the city's many bayous out of their banks and created flooding in parts of the city that had not flooded for many years, Turner said.
Flash flood warnings
were up in about two dozen Texas counties across in the southern part of the state, including the Houston and Austin metropolitan areas.
The situation is the result of a nearly stationary area of low pressure that has stalled over the western United States, allowing moisture from the Gulf of Mexico to flow into Texas over the last few days, according to CNN meteorologist Sean Morris.
Very heavy rainfall is expected to continue through Tuesday before the system begins to move off to the northeast and weaken, he said.