The League One outfit appealed against the initial decision due to mistaken identity, but they also felt that the transferred red card would be wrongful dismissal.
Despite the club providing a statement from Port Vale's Antony Griffith stating that Garner had not hit him during the 2-0 win for Preston, the Football Association have upheld the ban.
A statement on the club's official website said: "The FA decision to ban Joe Garner is a complete shock given the evidence. The match referee and the assistant referee nearest to the incident say that they did not see anything.
"The assistant referee furthest away stated that he saw Neil Kilkenny commit an offence that he clearly did not commit. The Port Vale player involved in the incident put in writing that at no stage did Joe Garner strike him. We are therefore dumbfounded as to what basis the FA can decide to ban Joe Garner."
Garner will miss games against Bristol City, Wycombe Wanderers and Crawley Town.