Police said three police officers were also injured in the incident which began with a traffic stop.
The suspect was shot dead in an exchange of fire with officers.
The shooting comes less than a month after a gunman killed 22 people in the Texas city of El Paso.
Some of the shots were fired on the Interstate 20 Highway linking Odessa and Midland, where cars were left with bullet holes.
President Donald Trump tweeted that he had been briefed by Attorney General Bill Barr.