Border Patrol agents were notified to watch for the truck that later plunged over the cliff.
A pickup hauling at least 18 people plunged off a more than 40-foot embankment and came to rest upside down south of Alice.
Patrol car dash cam video showed the rescue efforts.
Martinez said numerous responders Tuesday afternoon assisted the Border Patrol because "there were so many victims." Authorities said at least 18 people, suspected to be illegal immigrants, were in the truck and all were seriously injured.
Martinez said the Border Patrol had been notified by officers in another county to watch for the pickup, which then tried to flee.
Messages left Wednesday with a Border Patrol spokesman were not immediately returned to The Associated Press.
Alice is 40 miles west of Corpus Christi.