The first openly gay man to officiate at the highest level in the game was giving evidence to the Culture, Media and Sport Committee's inquiry into homophobia in sport.
Mr Owens advised youngsters receiving abuse to either report it, tell their parents or "try and share that abuse".
But he agreed that was often not easy.
Explaining his own own experience of being bullied, as a 12-year-old, Mr Owens said: "You feel it's your own fault. You feel a sense of weakness.
"You feel a sense of shame and embarrassment that you are a weak person because you are not able to deal with this.
"You're not the weak person, the weak person are the people who are doing the bullying and the abuse."
Mr Owens said that rugby was not a homophobic sport and that his experience in the sport had been hugely positive.
He said that there were some issues, but when rugby was aware of problems "it does all it can to eradicate them".
"When you have people chanting homophobic abuse then there has to be zero tolerance in the way that they deal with those people," he added.
A zero tolerance policy is needed on homophobic online bullying, international rugby referee Nigel Owens has told MPs.