The official identified him as Paul Ciancia. A second law enforcement official confirmed the identity, speaking on condition of anonymity.
Authorities say the suspect killed a TSA employee in the shooting.
Los Angeles police Assistant Chief Earl Paysinger, who would not identify the suspect, said he was apparently injured following an exchange of gunfire with officers, was in custody.
Tim Kauffman, a spokesman for the American Federation of Government Employees in Washington, confirmed Friday that a TSA officer was killed. Kauffman said the union's information comes from its local officials in Los Angeles.
The shooting started at 9:20 a.m. in Terminal 3, when a man walked in and pulled out an assault rifle from a bag and began opening fire, according to Patrick Gannon, chief of Airport Police at LAX. The shooter walked into the screening area and continued to shoot. He then continued into the airport, continuing to fire.
Officers tracked the shooter, exchanging fire with him. They were eventually able to take him into custody, Gannon said.
"This was a lone shooter. He was the only person armed in this incident," Gannon said. "There is a tremendous amount of investigative work that needs to be done."
Seven people total were injured, according to Jim Featherstone, general manager of the L.A. Emergency Management Department, though he would not say how people were injured or elaborate on the extent of the injuries during an afternoon press conference. Six people were transported to area hospitals, he said.
Three male patients were taken to Ronald Reagan University of California at Los Angeles Medical Center.
One was in critical condition and two were in fair condition at 1:30 p.m. Utah time, according to Lynne McCullough, UCLA emergency medicine physician.
The critical patient and one fair patient suffered gunshot wounds, she said. She did not provide further information, citing privacy concerns.
The Federal Bureau of Investigation is leading the investigation, and Special Agent David Bowdich said there were "no additional threats" as of 12:30 p.m. Utah time.
He would not provide further details.
Gannon said the shooting caused mayhem at the airport.
"As you can imagine, a large amount of chaos took place in this entire incident," he said.