The Uruguayan thanked a follower on his Instagram story last month using the phrase "gracias negrito," which translates to "thank you, little Black one."
The phrase is seen as an affectionate term of endearment in South America but considered offensive in other parts of the world, because its meaning can be misconstrued.
Cavani later issued an apology for the language used saying "the last thing I wanted to do was cause offence to anyone. I am completely opposed to racism and deleted the message as soon as it was explained that it can be interpreted differently."
In a statement on Thursday the FA said the post was "insulting, abusive, improper and brought the game into disrepute" and constituted an "aggravated breach [...] as it included reference, whether express or implied, to colour and/or race and/or ethnic origin."
Cavani must also complete "face-to-face education."
In a statement responding to the three-match ban, Manchester United said: "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."
The ban comes into immediate effect meaning he will miss United's English Premier League game against Aston Villa on Friday, the English League Cup semifinal against Manchester City on January 6 and the FA Cup third round tie against Watford on January 9.
Cavani has scored four goals since joining United in October.
Before moving to England, the 33-year-old Cavani won six league titles with Paris Saint-Germain and, with 200 goals, is the French club's all-time leading scorer.