Most of the vehicle's passengers were ejected.
The department originally reported that 10 people were killed but later lowered the number after learning two people were in the hospital with life-threatening injuries, not dead.
Authorities said all the victims are believed to be illegal immigrants from El Salvador, Guatemala, Honduras, Ecuador and perhaps Mexico.
"There was no rear seat in the (Ford) Excursion. They stack live people, as many as they can, like stacking wood," Craig told The Associated Press.
Five survivors were airlifted to a Tucson hospital and 14 were transported to hospitals in Tucson and Sierra Vista. Eight men treated and released from the hospitals were in the custody of the U.S. Border Patrol, agency spokesman Omar Candelaria said.
The identities of the driver, passengers and the vehicle's owner were being withheld pending their identification and notification of relatives. Many victims did not have identification, Craig said.
The department said the cause of the rollover is under investigation.
Investigators have not identified the driver, but if he survives, Craig said charges against him would be pending.
Craig said the remote area where the accident occurred is a route commonly used by those smuggling illegal immigrants into the United States.
