The former prime minister allegedly ordered that the unexplained incident over the east coast of England should be kept secret for at least 50 years because it would provoke "mass panic".
The claim, made in 1999 by a scientist who said his grandfather was one of Churchill's bodyguards, is recounted in declassified Ministry of Defence UFO files made available online by the National Archives.
The man, who is not named in the files, said Churchill was reported to have exclaimed: "This event should be immediately classified since it would create mass panic amongst the general population and destroy one's belief in the church."
Other incidents recorded in the 18 MoD files, which cover the years 1995 to 2003, include the famous 1974 UFO sighting over the Berwyn Mountains in North Wales, which officials concluded could be explained by a combination of an earthquake and a meteor.
The files also reveal a man went to RAF Stanmore in north London to warn military police he had dreamed that a bomb would explode in London in the "near future". About eight to nine weeks later, in May 1990, RAF Stanmore was targeted by a terrorist bombing.
They include a memo revealing that RAF jets were scrambled to investigate unexpected objects picked up on radar 200 times a year during the Cold War - but that this fell to zero after the collapse of the Soviet Union in 1991.
Nick Pope, a former MoD civil servant who worked on the official UFO files, said: "Whatever you believe about UFOs, there's some fascinating material in these real life X-Files.
"Most of these sightings turned out to be misidentifications of things like aircraft lights or meteors, but a small proportion could not be explained."
The files can be accessed at www.nationalarchives/ufos and will be free to view for the first month.
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