Video of the incident shows the vehicle already upside down on the ground, with several individuals crawling out of a side window before running away. Officers' vehicles can be seen at the scene, which was in Sullivan County, Texas, in the Rio Grande Valley.
Of the 11 migrants involved, seven escaped, including the smuggler, Texas DPS told Fox News.
The incident comes amid a surge of migrants trying to cross the southern border in numbers that are only expected to increase if President Biden lifts the Title 42 public health order later this month as intended.
A day earlier, Border Patrol officers in Texas seized $18 million worth of methamphetamine that was being smuggled in a semi-trailer truck from Mexico over the World Trade Bridge into Laredo.