By Agence France-Presse
The crash took place in the city of Big Wells outside San Antonio after Border Patrol agents tried to stop the driver of an SUV, ABC television affiliate KSAT reported.
A total of 14 people were in the vehicle, including the driver, who was not seriously hurt.
It quoted Dimmit County Sheriff Marion Boyd as saying the driver was traveling at speeds of 100 miles (160 kilometers) per hour while trying to flee from deputies.
While traveling at top speed, the driver lost control of the SUV, which rolled, causing the deaths and injuries, the Texas Department of Public Safety was quoted as saying.
The SUV driver was arrested by Homeland Security agents, KSAT quoted officials as saying. Four of the passengers were pronounced dead at the scene and another died at hospital. The injured were taken to hospitals for treatment.
The deadly crash comes as the United States grapples with a swirling crisis over illegal immigration that has seen nearly 2,000 children separated from their parents over a recent six-week period.