Justin Lee Price, who is 19, targeted Rashford with abuse on Twitter after the Manchester United player missed a penalty in the shootout won by Italy at Wembley Stadium in July.
Price previously admitted to one count of sending a grossly offensive message by public communication network at a hearing at Worcester Magistrates' Court on March 17.
Price initially tried to avoid detection by changing his Twitter username after the post was reported, British prosecutors said.
He then denied the offense in his first police interview following his arrest, but later admitted to posting the tweet when he was questioned by officers a second time.
He was sentenced at Kidderminster Magistrates' Court in Worcester.
"Price targeted a footballer based on the color of his skin and his action was clearly racist and a hate crime," said Mark Johnson of the Crown Prosecution Service.
"Those who racially abuse footballers ruin the game for all. I hope this case sends out the message that we will not tolerate racism and offenders will be prosecuted to the full extent of the law.''
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