Boyd said the Suburban was being chased by a Dimmit County deputy.
It was unclear why law enforcement began chasing the vehicle, though KABB quoted Dimmit County Sheriff Marion Boyd saying it was the result of "good police work".
The SUV carrying 14 people went out of control at more than 100 miles per hour and overturned on Texas Highway 85, ejecting most of the occupants, Dimmit County Sheriff Marion Boyd said.
One of the four taken to the San Antonio hospital later died, the department said. During a press conference, Boy said 14 people were in the vehicle, including the driver and passenger.More news: Who's the Spider-Man Game's Mystery Villain?
The border agent stopped one of the vehicles and another agent stopped a second one.
"From what we can tell the vehicle ran off the road and caught gravel, and then tried to re-correct and then caused the vehicle to turn over several times", Mr Boyd told CNN.
Border Patrol agents attempted to stop the vehicle when the chase ensued and were soon joined by Dimmit County deputies, KFDM reported. Authorities believe the driver and one passenger are U.S. citizens and the 12 other passengers were undocumented immigrants, Boyd said.
Some injured were taken by helicopter to San Antonio, about 90 miles (144.83 kilometres) northeast.More news: Kawhi Leonard wants out from San Antonio Spurs
Border patrol agents use all-terrain vehicles (ATV) to search for illegal immigrants along the US.
Boyd said the driver was known to police, and had been seen in the area "last week", either scouting the area or looking for immigrants to smuggle. "I think we need a wall in my opinion". All of the people killed were immigrants.
The sheriff said he didn't know where the immigrants were from but said most migrants who travel through the area are from Central America or Mexico.More news: Spieth, McIlroy among big names to miss US Open cut
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