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.
Los Angeles resident Jennifer Chan was in Terminal 3 at the time of the shooting, answering work emails when she saw people start running.
"It was pure pandemonium," she said. "I heard one gunshot as I was told to go down the stairwell."
She was stopped as she approached baggage claim, and police officers began streaming into the airport, she said. Officers told everyone to get down on the ground as they streamed past.
Chan caught several glimpses of the bedlam a little girl holding her dad's hand as she screamed and cried; an adult daughter asking her elderly mother "Are you OK. Can you walk? Can you do this?" as people ran by; a woman screaming "Where is my husband?"
Chan ended up at the nearby Radisson hotel, but all of her belongings were still at the airport Friday afternoon.
At about 1:30 p.m. Utah time, a large throng of air travelers had gathered about a half mile outside of the airport, attempting to get into the airport to catch their non-canceled flights out of other terminals.
Gina Marie Lindsey, executive director of Los Angeles World of Airports, said flights would be leaving only from the south end of the airport and at about half their normal rate.
"It will be a carefully orchestrated logistics ballet to get people rescreened," she said.
For the most updated information, Lindsey told air travelers to follow @LAX_Official on Twitter.
The Federal Aviation Administration issued a ground stop for all flights that were scheduled to land at LAX, according to Dave Korzep, superintendent of airport operations for the Salt Lake Department of Airports.
That means that any flight that has not yet left for Los Angeles will not be able to take off from its airport of origin, he said. Anyone scheduled to leave for Los Angeles should call their airline to determine the status of their flight.
"This could be a long-term event," Korzep said.
At the Salt Lake City International Airport, the Airport Police and TSA have been alerted about the LAX situation and are monitoring it. Officers have stepped up patrols throughout the airport as a precaution, Korzep said.
LAX air traffic controller Michael Foote said some flights were still being allowed to depart.
Foote said his colleagues in the control tower saw passengers spilling from the terminal onto the tarmac, "evacuating the building, getting out as fast as they could." Officers eventually corralled them.
The incident may impact the Utah State University football game scheduled for 2 p.m. Saturday as the University of Hawaii football team was at LAX Friday morning awaiting a charter flight to Ogden, according to the Honolulu Star-Advertiser.
"We don't know yet how this is going to impact us," Derek Inouchi, UH Sports Information director, told the Star-Advertiser. "We're just playing it by ear. We're watching it on TV like everybody else."
Terminal 3 is home to Allegiant Air, Frontier, JetBlue, Spirit, Virgin America and Virgin Australia, according to the LAX website.
Tribune reporter Matt Piper and The Associated Press contributed to this report.
Celebrities caught up in LAX shooting
LAX is the international hub for stars in and out of Hollywood, and a few celebrities were among the travelers caught up in the chaos following the deadly shooting at the airport Friday morning.
With time on their hands, they took to Twitter to update fans.
"At (hash)lax Some (expletive) shot up the place." James Franco, who included a self-portrait in an airplane window seat. His publicist confirmed the actor was a passenger on a flight that landed Friday morning after the shooting occurred.
"Currently waiting on the plane in a remote parking area. I am safe. Praying for the victims of this shooting." Nick Jonas, who posted moments earlier that the flight he arrived on was set to deplane at the terminal where the shooting occurred.
"Heard gun shots then everyone starting running for the door. Not sure if anyone was hurt. (hash)LAX" "Mythbusters" host Tory Belleci posted Friday morning, followed by a series of tweets and photos updating the situation at the airport throughout the afternoon.
"Almost 4 hrs since the shooting. Still here. 2000+ people. They're handing out waters & snacks. (hash)LAXShooting" Belleci. Discovery confirmed in a statement that he and Grant Imahara were at terminal 3 on route to Delaware when the shooting occurred.
"Otw to lax and I hear there was a shooting at the terminal I'm going to. No Dallas for me this weekend, hope everyone is ok. I hate u LAX" "Glee" star Mark Salling.
The Associated Press