The incident began about 6 p.m. when the driver "broke through a fence at the FedEx cargo facility and drove westbound on the surface road and airfield," according to an LAX statement.
Airport police then cornered the vehicle near the Atlantic FBO, a private terminal at the facility, and detained the suspect. He was not immediately identified.
News video from AIR7 HD showed the letters "SOS" spelled out on the hood of the suspect's car, apparently written with black electrical tape.
"Airport police and operations have inspected the south airfield following police activity that briefly closed the south runways," LAX said on Twitter shortly before 7 p.m. "One south runway has reopened and the other will open shortly. The north runways continued operating normally."
Video showed the car racing across the tarmac as several police vehicles chased the driver.
No injuries were reported.
"We have responded with our partners and the situation is being collectively assessed," an FBI spokesperson told ABC7.
The incident caused three planes preparing to land to be waived off, but no other flights were impacted.
Authorities later said that no federal charges were expected, adding that local police were leading the investigation.
No weapons were recovered from the driver.