Donnie Fraser, 19, shouted a racial remark at 17-year-old defender Jordan Tapping during the game between Peterhead and East Stirlingshire on Saturday.
The East Stirlingshire player, who is still a schoolboy, was so upset that his manager substituted him 15 minutes before the match ended and he left the pitch in tears.
Police launched a probe and arrested Fraser at his work on Tuesday afternoon. He appeared at Peterhead Sheriff Court yesterday.
The first-time offender admitted shouting a racially offensive comment likely to incite public disorder at the Balmoor Stadium in Peterhead.
Fiscal depute Ruairidh McAlister said: "As the game was ongoing, the accused was heard by persons round about him to shout, 'F*** you, you black c***'. This was directed at Jordan Tapping, who is a player with East Stirlingshire football club."
The incident was not reported to police but officers launched a probe after reports in the media.
Fraser, of Peterhead, admitted the offence when interviewed by officers and said there had been no reason for him to commit the offence.
Defence lawyer Garry Sturrock said his client was ashamed.
He said Fraser was arrested at his work where he was employed as a labourer at a local fish factory.
The court heard Fraser had suffered from ADHD as a child and had anger management issues. Mr Sturrock said he often got frustrated when playing football with friends when he was on the losing side and was an avid Peterhead FC supporter.
He told the court his client had felt Tapping had carried out an aggressive tackle during the match and was unhappy the referee did not intervene. Mr Sturrock said other fans were telling the player to go back to his own country.
Mr Sturrock said: "He tells me that racial remarks are completely out of character for him. He is deeply ashamed and accepts the wrongness of his actions."
Sheriff Alison Stirling told Fraser he had committed a "terrible offence" and said he had behaved in an unimpressive way.
She ordered him to pay £300 compensation to Jordan Tapping and banned him from attending football matches for 12 months.
An East Stirlingshire spokesman said the conviction showed racist remarks were unacceptable. He added: "It is very much our hope there are no more incidents of this type at matches."
A spokesman for the SPFL said: "We're pleased that the police and courts have moved so quickly to identify and punish an individual in relation to Saturday's unacceptable incident at Peterhead.
"In light of this, we will again speak to both clubs and police about our own investigation - which continues - and next steps."