Officials warned the devastation from the storms - which set records in parts of Texas' low-lying Rio Grande Valley - was only starting to be understood.
On the US side, officials said at least three people were killed in Hidalgo County, where officials said more than 53cm of rain this week soaked the city of Harlingen.
"The bed is the only thing dry right now, because the sofas are soaked. Everything is soaked," said Jionni Ochoa, 46, from his home in Palm Valley, near Harlingen.
He said water started coming into their house the previous night and began pouring out of the electrical sockets. They turned off the power and tried to save as much as they could.
The Mexican state of Tamaulipas reported that an 83-year-old man drowned in Reynosa, which is across the border from McAllen, Texas.
The US Customs and Border Protection said the driver of a vehicle suspected of taking part in migrant smuggling tried crossing a flooded roadway in Hidalgo County and plunged into a canal.
The agency said the body of one person who drowned was recovered and another was missing. It was not immediately known if those were among the deaths reported by county officials.
In Alamo, a small Texas border city, crews responded to more than 100 water rescues, including people stranded in vehicles and trapped in homes, Fire Department Chief R.C. Flores said.
Dozens more rescues were made in nearby Weslaco, which was inundated with about 36cm of rain, according to Mayor Adrian Gonzalez.
"It's a historic rainstorm, and it's affecting all the valley, not just Weslaco," Gonzalez said.
Thousands of power outages were reported, and more than 20 school districts and college campuses cancelled classes. Valley International Airport in Harlingen was closed on Friday, and all flights were cancelled.
Up to 31cm of rain fell in parts of northeastern Mexico, according to Tamaulipas authorities.
Luis Gerardo González de la Fuente, state co-ordinator of emergencies, said the most affected city was Reynosa but conditions were also dangerous in the border cities of Rio Bravo, Miguel Aleman and part of Matamoros, south of Brownsville, Texas.
Some 640 military personnel were deployed in the area. Authorities said electricity was being restored as water levels fell.
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