The SUV went out of control at more than 100 miles per hour and overturned on Texas Highway 85, ejecting those inside, Dimmit County Sheriff Marion Boyd said.
Boyd said Border Patrol agents began chasing the SUV around noon and then Dimmit County Sheriff's deputy took over, CNN reported.
All of the people killed were immigrants.
Several people were arrested after another agent stopped the second vehicle, but the third vehicle didn't stop when an agent tried to pull it over, Customs and Border Protection said.
The driver of the vehicle is believed to be a USA citizen. The sheriff said there were 14 people in the vehicle and 12 of them were ejected.
High-speed chases aren't uncommon in some US states bordering Mexico, as smugglers and others attempt to flee checkpoints and outrun the Border Patrol.
Four people were killed at the scene and a fifth person died at a hospital in San Antonio, the sheriff said. Four people died at the scene, four were taken to San Antonio Medical Center for treatment, and six were taken to the Dimmit County Regional Medical Center, according to the department.
"This is not unusual", Boyd said.
The chase started when a border control agent spotted three vehicles traveling together and became suspicious of smuggling. He said the pursuit was the result of "good police work", and gave his support for the construction of a border wall. "This is a flawless example of why our borders need to be secure", he added.
Boy believes the driver was scouting the area to try and pick up immigrants.
Border patrol agents use all-terrain vehicles (ATV) to search for illegal immigrants along the US.