"Improbable as that may seem to those of us of an older generation, we do not reject that assertion as untrue", read the statement.
The goalkeeper had admitted "from the outset" of the hearing that he did not know what a Nazi salute was.
An FA investigation into whether the picture, posted to social media and showing the Wales worldwide giving the straight-arm salute, revealed shocking details about Hennessey's lack of historical knowledge.
As part of his defence, Hennessey provided photographs which showed him using the gesture during matches as a way of communicating with team-mates on the pitch.
It said the FA was "entirely justified" in bringing the case but that "rather than giving a Nazi salute, we think it more likely that Mr Hennessey was, as he says, trying to shout at and to catch the attention of the waiter".
The written reasons for the decision, published on Tuesday, stated the 32-year-old was not punished because he proved he was unaware of the historical significance of the gesture.
Crystal Palace keeper Wayne Hennessey was criticised by the FA for his ignorance.
Palace also conducted an internal investigation, in which they also cleared Hennessey of wrongdoing - although the FA report scathingly comments this was 'an investigation which did not impress us'.
The commission was told there was no CCTV footage but heard witness statements in person from Hennessey and the waiter, and from Connor Wickham - whom the Wales worldwide, who had not been drinking, had later driven home - Meyer and Palace's player liaison officer Danny Young, who had been at the goalkeeper's side, via video link. Bussolini had apparently been asked to take a number of photos on various smartphones, holding the camera above his head and, at one point, standing on a chair to capture the whole table of Palace players. Hennessey said the resemblance was completely coincidental.