The incident occurred in the dying minutes of the Brewers’ 2-1 defeat against the Bees at Griffin Park on December 10.
The FA said in a statement: “Ben Turner will serve a five-match ban with immediate effect after an Independent Regulatory Commission hearing found the Burton Albion defender guilty of breaching FA Rule E3.
“The commission heard the allegation that in or around the 94th minute of the match between Brentford and Burton on Saturday 10 December 2016, Turner used abusive and/or insulting words towards Nico Yennaris, contrary to Rule E3(1).
“It was further alleged that this breach of Rule E3(1) was an “Aggravated Breach” as defined in Rule E3(2), as it included a reference to ethnic origin and/or race and/or nationality.
“As well as a suspension, the commission also ruled Turner must attend an education course and has been fined £8,000.”