The 33-year-old Uruguayan made the post on social media after he scored a brace in United's 3-2 comeback win over Southampton at St Mary's.
Cavani, who would later delete the post, was charged by the Football Association for using insulting and/or improper words.
The former Paris Saint-Germain and Napoli star pleaded guilty to the FA charge and will now serve a three-match ban, as well as being hit with a £100,000 fine.
Cavani will also be required to complete a "face-to-face" education programme alongside the ban and fine.
The United striker, who arrived on deadline day as a free signing, apologised over his message and insisted it was "an expression of affection towards a friend."
Cavani wrote in his statement: "Hello everyone, I do not want to extend much in this uncomfortable moment for me.
"I want to share with you that I accept the disciplinary sanction knowing that I am foreign to English language customs, but I do not share the point of view.
"I apologise if I offended someone with an expression of affection towards a friend, nothing further was my intention.
"Those who know me know that my effort is to always seek the simplest joy and friendship!
"I appreciate the countless expressions of support and affection.
"My heart is at peace because I know that I always expressed myself with affection according to my culture and way of life. I send you a sincere hug."
According to reports, the FA had charged Cavani for the post on 17th December.
Following Cavani's punishment, United said in a statement: "Despite his honest belief that he was simply sending an affectionate thank you in response to a congratulatory message from a close friend, he chose not to contest the charge out of respect for, and solidarity with, the FA and the fight against racism in football.
"The club trusts that the independent regulatory commission will make it clear in its written reasons that Edinson Cavani is not a racist, nor was there any racist intent in relation to his post."
Cavani will now miss United's New Year's Day Premier League clash with Aston Villa, as well as their League Cup semi-final match against Manchester City and FA Cup tie against Watford.
Featured Image Credit: PA
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