The Uruguayan thanked a follower 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.
Having come on as a substitute at halftime on Sunday with Manchester United 2-0 behind, Cavani had an instant impact, scoring twice and assisting the other goal to help his side to a dramatic victory.
"The message I posted after the match on Sunday was intended as an affectionate greeting to a friend, thanking him for his congratulations after the game," Cavani said in an apology on the Manchester United website. "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. I would like to sincerely apologise for this."
Manchester United followed up his statement with one of their own, saying: "It is clear to us that there was absolutely no malicious intent behind Edinson's message and he deleted it as soon as he was informed that it could be misconstrued. Edinson has issued an apology for any unintentional offence caused. Manchester United and all of our players are fully committed to the fight against racism."
Before Cavani released the apology, England's Football Association confirmed to CNN it was aware of the social media post and was looking into it. According to new FA regulations, any player found guilty of an offense of discrimination will face a minimum six-match suspension. But if a regulatory commission finds there was no "genuine intent" to discriminate or cause offense, the suspension could be dropped to three games.
Comparisons are being drawn with a 2011 incident involving another Uruguayan player, Luis Suárez.
Suarez, then a striker for Liverpool, repeatedly called then-Man United captain Patrice Evra "negro" during a Premier League game.
At the time, Suarez argued the term was not offensive in his homeland. The FA, however, disagreed with Suarez. A report into the incident said that Suarez did not use the term "negro" in "the conciliatory and friendly way that was common and inoffensive in Uruguay."
Suarez was banned for eight games as a result and fined 40,000 pounds.