Plasterer Bradford Pretty, 50, embarked on a drunken tirade against the three England players following the loss on penalties to Italy in July.
The clip – which includes him using two racist terms to describe the three young men – was uploaded to social media and attracted comments.
Pretty appeared at Folkestone Magistrates’ Court on Wednesday where he admitted sending a message that was grossly offensive or of an indecent, obscene or menacing character.
He was sentenced to 50 days in prison, suspended for 12 months, and ordered to complete 200 hours of unpaid work.
Playing the video for the court, prosecutor Julie Farbrace said: “It shows him talking about the game… in particular talking about the England players who had missed a penalty at the final.
“In relation to the matter there were people who commented on the video, people who were upset about the word he put in there.”
In the video, an intoxicated Pretty can be heard saying: “Where do I start? Where do I start?
“So gutted like all of us.
“Proper deflated, big proud of the boys, big proud, but anyone and everyone that knows me well will understand what I am talking about.”
He goes on to refer to Rashford, Sancho and Saka missing penalties and uses two racist terms to refer to them.
“I am so f****** sad. I am so f****** sad,” he adds at the end.
When challenged in the comments section of the video, Pretty apologised to anyone that was offended but railed against “political correctness”.
He added: “I am standing up and saying what I said for the weak ones…
“England till I die.”
His defence solicitor Richard Graham acknowledged the video was “abhorrent”, saying it was a “moment of drunken madness”.
He said Pretty had drank “15 or 16” cans of lager on the day of the final and was “clearly heavily intoxicated” in the video.
Mr Graham asked that his client be given credit for his early guilty plea.
Chairman of the bench Chris Mackenny told Pretty that his offending “clearly passes the custody threshold” but suspended the sentence of 50 days in prison for 12 months.
Pretty, of Folkestone in Kent, was told to complete 200 hours of unpaid work and must also pay prosecution costs and a court surcharge totalling £213.
England lost 3-2 on penalties to Italy after a 1-1 draw at Wembley in the final on July 11 this year. Rashford, Sancho and Saka all missed from the spot for England in the deciding shoot-out.