Trapp was walking pet bull terrier Trigger in West Ewell, Surrey, on May 6 last year when he got into a row with Mr Matangambiri.
Guildford Crown Court heard how Mr Matangambiri had confronted 41-year-old Trapp after he called him a 'n*****' as he walked along the street.
As the pair rowed, however, Trapp let his dog off the leash and ordered it to bite Mr Matangambiri, who had to be treated in hospital for a number of dog bites.
Trapp claimed in court that he was shouting the dog's name, Trigger, but the judge rejected his claims.
The court heard how after being arrested for the race attack, Trapp went to the victim's home with a group of friends to threaten him and order him to withdraw the claim against him.
Trapp eventually ended up forging a letter from Mr Matangambiri and sending it to police, but was caught out when they realised he had spelled the victim's name wrong, the court heard. They did not disclose how he had spelled the name.
Leesha Whawell, prosecuting, told the court: "The police officer in the case became suspicious because the complainant's name was mis-spelt."
Clodaghmuire Callinan, defending, said that Trapp had not said 'the N word', but was shouting his dog's name, adding: "Mr Matangambiri mistakenly thought he was shouting the word and an argument developed from there."
Judge Michael Addison, in jailing Trapp for eight months, rejected the defence, telling him: "Racist behaviour will not be overlooked and tolerated.
"The dog was very aggressive and you knew it."
Trapp, of Tadworth, Surrey, admitted racially-aggravated common assault, witness intimidation and having a dog dangerously out of control in a public place.
The judge ordered that the dog be put down.