Jorge Vargas, 27, was rushed to a hospital after the 4 a.m. attack and died a short time later, Phoenix police Sgt. Trent Crump said.
No one was in custody and there was no information on a motive.
"It's really a whodunit," Crump said, adding that investigators are looking into whether the shooting was connected to the victim's job.
"Anytime we have a homicide and the suspect is unknown, we start with our victim," Crump said. "But we can't rule out a random act."
Vargas was an eight-year veteran who worked for the Sheriff's Office Custody Support Bureau in the jail's food factory, but he often came into contact with inmates, Sheriff Joe Arpaio said.
He wouldn't speculate about whether Vargas' killing may have been related to his work, but said his office was assisting Phoenix police in the investigation, including gathering intelligence from the jails.
"I won't call it an assassination or an execution," the sheriff said. "Right now, I'll call it a murder."
Arpaio did, however, have a warning for the killer.
"Watch out," he said. "We're ready for battle and we're going to take whatever action we need."
Vargas was wearing his uniform that identified him as a Sheriff's Office employee at the time of the attack and was in his driveway working on his car when he was shot.
A man was seen running from the scene after neighbors heard shots and found the critically wounded officer in his driveway.
Tactical teams and K-9 units searched the area, but so far police had no suspects.
Associated Press writer Paul Davenport contributed to this report.
Copyright 2013 The Associated Press.