Airport spokesman Joe Rajsel told CNN that Las Vegas Metropolitan Police and airport officials had taken the man into custody and were investigating.
The man threw down a perimeter fence and was spotted by airport staff at around 1.40pm local time, Mr Rajsel said.
Video captured by a passenger shows the person sitting and walking on the wing.
He would then remove his socks and shoes before attempting to board the Boeing 737’s winglet, which turned the wing of the aircraft upside down.
When police officers approach the person, he slides the winglet down and falls into the dorm below.
Authorities took the man into custody and he was taken to a medical facility, Mr Rajsel said.
Alaska Airlines reported that the plane returned to Gate for a full inspection following the incident.
According to the Alaska Airlines website, the flight was scheduled to depart at 12.30pm, but the delay led to the departure at 4.48pm.