A court heard Martyn Howlett was walking down Muller Road in Horfield after watching Rovers play Norwich City with his family and friends when he made his shocking outburst.
Bristol Magistrates' Court was told that the 43-year-old gestured at a British National Party badge he was wearing as he abused a group of Asian children having a picnic outside the 1st Bristol Muslim Scout group's headquarters.
Waving his arms in a Nazi salute and pointing to his badge, he shouted: "It's BNP, racist, fascism and Hitler", before launching into a foul-mouthed tirade in front of the frightened children.
The court heard he then directed abuse at a young girl wearing a head scarf.
Prosecuting, Michael Hartnell said: "He was making hand gestures towards the children, and said they were out to blow something up. The children were scared and extremely concerned for their safety."
When asked what was wrong by Scout leader Sohail Javaid, Howlett repeatedly said: "This is our country, get out."
After police arrived, he said: "What is wrong with doing this? I am a right-wing extremist and I hate blacks."
The court heard Howlett, whose father Colin was the voice of Bristol Rovers on Radio Bristol for almost 30 years, had drunk about four-and-a-half pints of beer before the incident on May 1. On May 24 he was given a two-year conditional discharge after pleading guilty to committing racially aggravated disorderly behaviour.
Mr Hartnell told the court that Howlett, a Bristol Rovers season ticket holder for 37 years, had received a caution in 2008 for similar behaviour.
Howlett, 43, who lives in Muller Road, Eastville, defended himself in court.
Magistrates gave Howlett a three-year football ban, which prohibits him from entering an area 750m around the Memorial Stadium on Rovers match days for three hours before kick-off and two hours afterwards.
He is also banned from going to any football match in England or Wales, must surrender his passport to police before England away games, and not to travel to any towns or cities where Bristol Rovers or England are playing.
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