Hennessey denied the charge, taking to Twitter to explain that he had been calling out and waving to attract the attention of the person behind the camera when the picture was taken.
"I'm delighted that the FA have found me not guilty of this charge," Hennessey said in a statement on Palace's website.
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.
"This was a genuinely innocent moment, which appeared to be something completely different when captured on camera," Hennessey added.
"I want to state for the record that I abhor all forms of racism, fascism, anti-Semitism or discrimination of any kind."
(Reporting by Simon Jennings in Bengaluru; Editing by Toby Davis)
