In reasoning published on the FA website, the panel set out why the Wales international was cleared after being pictured with his right arm in the air in a photo posted on Instagram by his German team mate at Palace Max Meyer.
Hennessey had said he was simply raising his right arm to attract the attention of a waiter taking the photograph, while also cupping his left hand over his mouth to make his voice carry.
The photograph prompted an outcry on social media and the FA charged the 32-year-old with a breach of its rules relating to abusive, insulting or improper conduct and bringing the game into disrepute.
In his defense, the South London club produced further photographs showing the goalkeeper making similar gestures during matches while shouting at team mates.
"Mr Hennessey categorically denied that he was giving a Nazi salute," the written reason stated. "Indeed, from the outset he said that he did not even know what one was.
"Improbable as that may seem to those of us of an older generation, we do not reject that assertion as untrue.
"In fact, when cross-examined about this Mr Hennessey displayed a very considerable - one might even say lamentable - degree of ignorance about anything to do with Hitler, Fascism and the Nazi regime."
The report said that while such ignorance was regrettable, it did not mean the player was not telling the truth."All we would say (at the risk of sounding patronizing) is that Mr Hennessey would be well advised to familiarise himself with events which continue to have great significance to those who live in a free country," it added.