The hackers replaced the page's photo banner with the black and white flag of the Islamic State (IS) group, which is also known as ISIS, and filled its timeline with pictures declaring support for the jihadists and mocking Kim's regime.
One image showed Kim weeping at his father Kim Jong-Il's funeral in 2011, with the caption "crying pig".
The page's profile picture was changed to an image of a fighter with his face wrapped in a scarf, captioned "CyberCaliphate" and "I love you ISIS".
Another photo showing IS fighters was captioned: "North Korea the communist thug nation and the Chinese communist thugs will pay a price for their collaboration with the enemies of the muwahideen."
It is not confirmed that the Facebook page, which has some 8,000 "likes", is run by the airline.
South Korean news agency Yonhap described the page as Air Koryo's Facebook account and said its earlier posts had been deleted by the hackers.
A report by the website NK News in September said the page, which previously posted schedule information and news on aviation tours to North Korea, was operated by airline enthusiasts.
The hacking came two days after a group styling itself as the CyberCaliphate, using the same "I love you ISIS" profile picture, hacked US Central Command's social media accounts, forcing the military to take down the command's Twitter feed.
The websites of major North Korean propaganda outlets and the homepage of Air Koryo were blocked for days last month after US President Barack Obama pledged a "proportional" response to a massive cyber-attack against Sony Pictures.
The FBI has said North Korea carried out the Sony hacking in retaliation for the studio's comedy "The Interview", which centres on a plot to assassinate Kim, and the US government slapped new sanctions on Pyongyang this month.
North Korea has repeatedly denied involvement in the cyber-attack against Sony.