Sheriff Jim Winder called Joseph's weapon a "semi-automatic machine pistol."
Hoyal said the officers returned fire until the suspect dropped to the ground. When the officers approached the suspect to arrest him, they realized the man had not been hit by the gunfire, Hoyal said.
Joseph was expected to be booked into the Salt Lake County jail later Thursday night.
Hoyal said no officers were injured in the shooting, and no residents in the area were harmed by stray bullets. At least one bullet struck a parked car and police were working Thursday afternoon to identify other impact sites.
Kelly Andersen lives just around the corner from where the shooting happened. Andersen was at home Thursday afternoon when he heard police sirens. Moments later, he heard "five or six gunshots."
Andersen said he ran outside and saw a bald man in police custody. The man was sitting in the back of an ambulance and looked like he was high.
"He had his head down and looked all strung out," Andersen recalled.
Andersen moved to the neighborhood in mid-2013 and said he hadn't seen anything like the shooting before.
Police were investigating the amount of property damage caused in the shootout, but Hoyal believes it is minor.
Late Thursday, police flew a drone along the route of the suspect's path to obtain aerial shots for later crime scene reconstruction.
Court records indicate Joseph has an extensive criminal history dating back to the mid-1990s. His more serious past offenses include forcible sexual abuse and assault by a prisoner, court records show.