Anelka, playing for West Brom at the time, made the gesture having scored against West Ham on December 28. The striker was later sacked by his club following their own internal investigation.
The gesture, created by French political activist and comedian Dieudonne M'bala M'bala, was deemed by the Football Association to have an anti-Semitic element to it.
Assou-Ekotto was charged after sending a tweet to Anelka congratulating him on the gesture and, despite the 30-year-old's denial of wrongdoing, he will now serve a ban and pay a £50,000 fine.
The Cameroon international was also warned as to his future conduct, ordered to complete a compulsory education course and pay the full costs of the hearing.
Mr Peter Griffiths QC, the chairman of the Regulatory Commission, said in delivering the decision:
"Even though we have found that there was an aggravated breach of FA Rule E3 we are satisfied that when the player sent the tweet on the 28th December 2013 congratulating Anelka, in his mind he believed he was congratulating Anelka on what he perceived to be an anti-establishment gesture as opposed to one associated with anti-Semitism.
"But we are also satisfied of two further factors relevant to his culpability: That he was certainly aware before he sent the tweet that the quenelle gesture was very much associated with Dieudonne.
"And that he had, by then, acquired at least some knowledge of the controversies surrounding Dieudonne in the autumn of 2013 and that these had included, rightly or wrongly, allegations concerning anti-Semitism."
Assou-Ekotto said: "The FA will today release the findings of its Disciplinary Panel in relation to charges brought against me following a Tweet I posted in December 2013. I am obviously very pleased that the FA has accepted I am not an anti-Semite.
"I have always said that when I tweeted Nicholas Anelka I was congratulating him on an anti-establishment gesture and NOT on an anti-Semitic gesture.
"I am also therefore pleased that the Panel has found this to be true. The finding against me was that I breached the FA's rules but that my breach was unintentional. I have always loved the fans of Spurs and I want to assure them that I would never knowingly behave in a racist or anti-Semitic manner."