The boy was understood to be playing for Swindon Town AFC in an under-14s match when the incident happened.
Wiltshire FA handed him a two-year suspension from play and said he would be sent on a temper management course.
FA county secretary Mike Benson said the behaviour of some Premiership stars was being copied at junior level.
'Isolated incident'
"If you look into the trends, what happened in football on the television - within a week or two afterwards the same thing is happening in a local junior park, " he said.
""Television has a responsibility to think about how they show these things"
"This is the problem when role models don't set a good example in the professional game, that feeds down to junior footballers.
"Television has a responsibility to think about how they show these things."
He added that while the 13-year-old's behaviour was "an isolated incident" in that age group, more and more violence was beginning to filter into junior football.
"That's the first one we've had to deal with at that particular age but the problem is increasing in junior football."
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