The 33-year-old Uruguayan used the word "negrito" in an Instagram post last month after beating Southampton on November 29, before taking it down and apologising.
Cavani will miss United's league game against Aston Villa, the League Cup semi-final against Manchester City and the FA Cup match against Watford as a result of the suspension.
The FA said Cavani would also have to complete "face-to-face education" as part of his punishment.
United said Cavani chose not to contest the charge out of respect for the FA and the "fight against racism in football".
