Emergency crews juggled an onslaught of distress calls, but there were no immediate reports of any serious injuries.
"We're doing OK at the moment," said police spokesman Javier Sambrano.
Authorities said at least 60 people had been rescued, some standing on roofs, others atop cars.
Fire officials said they were worried about mudslides, boulders and other debris falling from rocky cliffs around several El Paso neighborhoods.
Rosa Reyes was given five minutes to evacuate with her 6-year-old daughter and a neighboring family after a rock wall behind her home collapsed. The tumbling wall punched a hole in the side of a house.
"The material things can be replaced," she said. "It sure didn't feel like five minutes."
A small apartment complex on a hill above Reyes' home was in danger of collapsing and had to be evacuated, said fire Capt. Keith Burch.
The rain threatened to push the Rio Grande over its banks and more than doubled the normal speed of the river's current, authorities said. The river started to recede Tuesday night.
The parched region had less than an inch of rain in the first six months of the year. But it may have gotten as much as six inches since Sunday, the National Weather Service said. Forecasters expect the rain to continue through at least Wednesday morning.
Volunteer fire departments around the county helped residents sandbag their homes.
In the city of Socorro southeast of El Paso, 60 National Guard troops were deployed to help residents, said state Sen. Eliot Shapleigh. The troops were borrowed from Operation Jumpstart, President Bush's initiative to use troops to combat illegal immigration.
The situation in Mexico was equally bad Tuesday, prompting Mayor Hector Murguia Lardizabal to declare a state of emergency in Juarez.
Fearing that the Rio Grande would overflow, authorities started evacuating more than 1,000 families from low-lying areas of the city.