Sgt. Steve Martos, a police department spokesman, said one of the victims called 911 at about 9 p.m. Wednesday. Officers who responded to the Mater Misericordiae or Mother of Mercy Mission found the two wounded priests.
One of the men died a short time later at a hospital.
A local television reporter, Chris Williams, discovered a car about four blocks from the church that had apparently been abandoned by the burglars. Martos said the green Mazda had been stolen at the church.
Martos said no arrests had been made early Thursday.
Mater Misericordiae, near the Arizona capitol, says on its website that it is a center for the traditional Roman Catholic liturgy, including the Latin mass.