Officers arrested Pete Garcia Jr., 55, at a Cherry Street residence after receiving an anonymous phone call, said Police Chief Toney Cowan.
Garcia was arraigned for murder. Justice of the Peace Linda Canon set bond at $1 million.
Garcia, who lives on Holly Street, was taken into custody without incident. He was booked into the Hockley County Jail at 8:39 p.m.
Lubbock police received a call at about 6 p.m. Sunday regarding shots fired at St. Elizabeth’s Catholic Church in the 2000 block of Broadway.
Officers searched the building but found no shooter, said Assistant Police Chief Jon Caspell.
Garza, 51, was shot in a parking lot after he and his girlfriend left a church service.
‘It was a targeted shooting,’ Caspell said.
The victim was taken to University Medical Center where he died at about 7 p.m.
A murder warrant was issued at 11 p.m. Sunday for Garciaís arrest.
Caspell said investigators donít have a clear motive for the murder.
But the two men apparently knew one another.
Garzaís girlfriend, Judy Garcia, was the ex-wife of Pete Garcia Jr.
Investigators believe a handgun may have been used. Caspell said he did not know how many times Garza was shot.
Garcia was transferred from Levelland to Lubbock Monday morning.