Cavani was sanctioned after replying to a friend on Instagram, using the Spanish term "Gracias negrito" -- which can be interpreted as racist in the United Kingdom.
Although it is widely used as a term of endearment in South America and the Caribbean, the word is regarded in England, as well as within parts of the Afro Uruguayan community, as carrying racial connotations.
"CONMEBOL express their solidarity with player Edinson Cavani, who was sanctioned by the English FA," a statement read. "The punishment handed to the said Uruguay international clearly does not take in account cultural characteristics and certain terms used in everyday life in Uruguay.
"The judgement on these types of declarations within the framework of a process which could lead to punishment for the athlete and could affect their reputation and good name, should always be carried out taking into account the context in which they were used and, above all, the cultural peculiarities of each player and each country.
"CONMEBOL condemns and will always condemn to the highest possible degree, any form of racist or discriminatory act, but the particular case for which Cavani was sanctioned does not form part of those."
The statement comes just one day after Uruguay internationals Diego Godin and Luis Suarez demanded the FA overturned the ban, and labelled English football's governing body "ignorant" and "racist."
Uruguay's Football Federation had also called on the FA to review its decision, while the Uruguayan Academy of Letters also denounced the suspension.