In a statement, the Border Patrol said the chase began after an agent spotted "three vehicles traveling in tandem pass his location on a rural highway" and the agent suspected that "a smuggling event was taking place".
Most of the occupants were believed to be living in the USA without legal permission.
"We've seen this many, many times, in not only this county but other counties along the border", Boyd said.
Four people were pronounced dead at the scene while a fifth died at a hospital.
It appears the SUV was being pursued by both Border Patrol agents and sheriff's deputies.
The Border Patrol said in a statement Sunday night that two other vehicles had been travelling alongside the SUV earlier in the day.
Four victims were dead at the scene, Boyd 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.
Border patrol agents use all-terrain vehicles (ATV) to search for illegal immigrants along the US.
KABB in San Antonio reported the driver was believed to be a USA citizen, and had been transported to a hospital.
Several people were ejected from the vehicle during the crash. The driver was not seriously injured and taken into Homeland Security custody, Dimmit County Sheriff Marion Boyd told reporters.
"This, I think, is a ideal example, of why are borders need to be secured", Boyd said. Authorities say the vehicle topped speeds of over 100 miles per hour.
The sheriff said he hadn't spoken to Border Patrol on the exact cause of the pursuit and nature of the crash.