Phillip Gannon, 58, was seen on live television doing a racially offensive monkey impression, Liverpool Magistrates' Court was told.
The incident happened amid a volatile atmosphere at Anfield on January 28 this year when the two sides met in an FA Cup fourth round clash, Rob Girvan, prosecuting, told the court.
It was the first game between the Premier League rivals since an eight-match ban was imposed on Liverpool striker Luis Suarez after the FA found him guilty of racially abusing Evra.
In the 23rd minute of the game, the TV camera panned over fans in the lower Centenary Stand, where Gannon was sitting with two friends, and he was seen to position his hands under his armpits.
The coverage, broadcast worldwide by ITV, quickly spread across social networking site Twitter and complaints were made to police, the court heard.
Gannon, a carpenter and joiner from Heol Meirion, Barmouth, north Wales, was found guilty of two counts of using abusive or insulting words within the hearing or sight of a person likely to be caused alarm or distress.
The bench imposed a four-year banning order which stops Gannon from attending Liverpool and England games and also bans him from Liverpool city centre four hours before and after a match.
Gannon was also fined a total £180 and ordered to pay prosecution costs of £300.