U.S. Customs and Border Protection said 18 people were in the SUV at the time of the accident in Vail.
Pima County Rural/Metro Fire Chief Willie Treatch reported a higher number of occupants in the SUV -- 22.
Treatch said there were five deaths at the scene and the injured were transported to a hospital, some by helicopter.
The Border Patrol tried to stop the SUV for an immigration check Saturday night, the Los Angeles Times reported.
The SUV did not stop and continued traveling west on Interstate 10, flipping over on the exit ramp for Highway 83, Customs and Border Protection officials said.
"Preliminary reports indicate that the driver of the Tahoe lost control on the exit ramp and rolled the vehicle, resulting in multiple fatalities and injuries," the agency said in a statement.