Anelka made the gesture - thought to have anti-Semitic connotations - after scoring for West Brom against West Ham in December last year.
The former France international denied the offence and defended the gesture by saying it was anti-establishment and a dedication to his friend and comedian Dieudonne M'bala M'bala
Anelka was subsequently given a five-match ban and sacked by his club, while Assou-Ekotto's tweet, which has since been deleted, read: "I congratulate you on the beautiful quenelle."
On Friday, the FA announced that the Tottenham man - who is not registered to play in the Premier League - has been handed a three-match suspension as well as a £50,000 fine.
The left-back may appeal the decision, though the FA did not disclose Assou-Ekotto's deadline which to do so.
A statement read: "The charge was that the comment was abusive and/or indecent and/or insulting and/or improper, contrary to Rule E3(1). Further to that, the FA considered the comment to be an aggravated breach, as defined in FA Rule E3(2), in that it included a reference to ethnic origin and/or race and/or religion or belief."
"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 December 28, 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," said chairman of the Regulatory Commission, Peter Griffiths QC.
"But we are also satisfied of two further factors relevant to his culpability:
1) That he was certainly aware before he sent the tweet that the quenelle gesture was very much associated with Dieudonne; and
2) 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."
