The vehicle was about 150 miles (241km) north-east of the US border when it careened off the highway and struck two trees on Sunday evening.
Immigration officials said that among the dead were citizens of Mexico, Guatemala and Honduras.
At least 23 people had been crammed inside the cab and bed.
"This is the most people I've seen in any passenger vehicle, and I've been an officer for 38 years," said Texas public safety department spokesman Gerald Bryant.
He told the Associated Press that several survivors with life-threatening injuries had been brought to nearby hospitals.
The driver was one of 11 who died at the scene.
US immigration officials were investigating whether it was a case of human-trafficking.
In April, nine Mexican immigrants died near the border after the driver of a minivan crashed while fleeing officials. That van had 18 people inside.
Fourteen people have died and nine were injured after a pick-up truck filled with illegal immigrants crashed in Texas, said the authorities.