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.

Phillip Smith, 40, was arrested in a hostel in Rio de Janeiro by Brazilian police. He is expected to be extradited or deported.
Smith used a passport in his birth name, Phillip Traynor, to fly to Chile and then go to Brazil on 6 November.
He escaped while on a three-day release from a prison south of Auckland.
Smith was sentenced to life imprisonment in 1996 after he murdered the father of a 13-year-old boy he had sexually abused between 1992 and 1995.
The family had moved house but he had tracked the boy to their new home in Wellington in December 1995.
He repeatedly stabbed the father, killing him. The boy managed to escaped and raise the alarm.
The victim, who is now 30, told New Zealand's Fairfax media after news of Smith's capture, that he could now stop sleeping with a knife under his pillow.
"It's great news, it's excellent," he said from his new home in Australia. "[I feel] relief, a bit of peace of mind that I can carry on with my life without having that fear over my shoulder."
Smith's escape has thrown Corrections and various other government agencies into controversy. It has led to calls for an overhaul of short-term prison release and the monitoring of high-risk offenders.
In an email Smith sent to Radio New Zealand he said he had planned the escape by running a criminal check on his birth name and discovering that it had not been red-flagged.
"My only anxiety was that somebody that knew me might happen to be at the airport at the same time to identify me,'' he wrote, according to Radio NZ.
He also said he had been able to fund the escape with money from various businesses he had run from prison.
Smith said he chose Brazil because he thought it would be more difficult to extradite him if he was caught, as the two countries do not have a formal extradition treaty.
He was arrested after a member of the public in Rio De Janeiro recognised him from images being circulated in hostels and tipped off police. Smith appeared in court after his arrest and has been remanded in custody for 60 days.
New Zealand Police Detective Superintendent Mike Pannett said in a press conference on Thursday that it was a relief to have Smith back in custody.
He would not, however, say how long he expected Smith to remain in Brazil.
"New Zealand will bring him back as soon as possible but we are reliant on the Brazilian justice system to process and deal with the situation," Mr Pannett said.
Criminal Bar Association President Tony Bouchier said police needed to be more thorough when checking the names of criminals.
"There's got to be a far more robust process gone through when police are entering people's names into their systems and starting somebody off with a criminal record," he told Radio NZ.
Corrections Minister Peseta Sam Lotu-Iiga urged a review of the incident "so we can make some decisions as to how we can improve how we have temporary releases done out in the community", according to Radio New Zealand.
A zero tolerance policy is needed on homophobic online bullying, international rugby referee Nigel Owens has told MPs.
[NEXT_CONCEPT]
A New Zealand man serving a life sentence for child molestation and murder who escaped to Brazil is now in custody, New Zealand police say.