Deputy William Coleman, 50, was struck moments after he and other officers arrived to investigate a report of a pre-dawn break-in at the building in the subdivision of Anthem.
"The suspect came out and started shooting at our deputies," Arpaio told reporters at John C. Lincoln Deer Valley Hospital.
The sheriff added that another deputy "returned fire and killed the suspect."
The Arizona Republic said Coleman was a 20-year veteran of the sheriff's department and was the first Maricopa County deputy killed in the line of duty since 1995.
The suspect was not immediately identified, but was reportedly toting a semi-automatic rifle that was too powerful for Coleman's bulletproof vest. Investigators seized the suspect's van as well as a pit bull waiting in the vehicle.