The City of Fallon Police Department identified the suspected gunman as John K. O'Connor.
O'Connor was a member of the congregation at the Church of Jesus Christ of Latter-day Saints on West Richards Street in Fallon and had attended church services prior to opening fire about 1 p.m. Sunday local time (4 p.m. ET), police said in a statement.
The gunman killed Charles E. Miller, known to Fallon residents as "Bert," and injured another person before walking home, police said.
"City of Fallon Police Officers and Churchill County Sheriff's Deputies surrounded the home while a hostage negotiator was called to the scene. After placing phone calls to the home, O'Connor agreed to exit the property and surrender to officers," the statement said.
'Does not appear ... directed at the church'
"It is too early to understand the motive, but initially, it does not appear to have been directed at the church, but an individual victim," the statement said.
O'Connor was being held at the Churchill County Jail and has yet to be formally charged, police said later Sunday.
The second shooting victim was treated at a hospital and later released.
At least 50 people witnessed the shooting inside the church, Fallon Chief of Police Kevin Gehman told a news conference. He said the gun believed to be used was a mid-size caliber handgun.
Police had "contacts" with O'Connor in the past but "nothing that would rise to this level," Gehman said.
Both Nevada's senators tweeted their condolences to the shooting victims.
Republican Sen. Dean Heller described the tragedy as "senseless."
"Deeply saddened by the act of senseless violence in Fallon today at the LDS church. We are in contact with Mayor Tedford and other officials. My family and I are praying for all loved ones, churchgoers, and community members impacted by this tragedy," he said.
Democratic Sen. Catherine Cortez Masto called for an end to gun violence.
"Devastated by the horrific news of the fatal shooting at the LDS church in Fallon. Thank you to the first responders on the scene," she tweeted. "My heart goes out to all of those impacted. Places of worship should be a safe haven. The gun violence across this country must end."