Having won the first leg 2-1, Arsenal endured a difficult night at the Camp Nou and Barca took a deserved lead at the end of the first half when an ill-advised Fabregas backheel enabled Andres Iniesta to free Lionel Messi.
Sergio Busquets then headed into his own goal to make it 1-1 on the night and 3-2 to the Gunners on aggregate, but their hopes were swiftly dealt a blow when Robin van Persie received a second yellow card for kicking the ball away. His shot had come just one second after the referee blew his whistle amid loud noise from the crowd.
Xavi then levelled the scores on aggregate before Messi converted a penalty to seal a 4-3 aggregate win.
After the match Cesc Fabregas wrote on Twitter: "Great support from the arsenal fans. I take full blame for the result tonight. One of the worst moments of my life. I apologise."
Fellow midfielders Samir Nasri and Jack Wilshere also took to Twitter to register their disappointment at the referee's performance.
In a tweet laced with irony, Wilshere wrote: "Arsenal fans great as always! Sorry about the result,good luck to Barca,great team! Oh yeah and the ref was good aswell..."
Nasri wrote: "Thank you for your support once again u were amazing sorry for the result now we have to think about Man utd game and we still have a league to win and by the way bussaca is the best."
Arsenal were second best even with 11 men on the field and, throughout the entire game, were unable to register a shot on goal and had only 31% possession.
However, Wenger believed there were signs at the start of the second half - when his team found the equaliser - that the game was turning and indicated that the Van Persie dismissal had made all the difference.
"I felt Barca gave a lot in the first half," he said. "We were completely dominated, I agree with that, but in the second half we felt there was more space. I knew that, like the first game, we could come back into the game and overall I'm convinced we would have won this game.
"I think two kinds of people will be unhappy - those who love Arsenal, and those who love the game. We are frustrated with his decision. It is very difficult to understand his attitude.
"We lost against a very good Barca side, congratulations to them and good luck for the future, but we have many regrets tonight because we didn't expect to lose the game like that. I feel sorry for the people who watched the game tonight."
Wenger confirmed he spoke with referee Busacca - and UEFA representatives - but said: "I think he must think that he made the right decision."
He added: "I just spoke to UEFA people. They are shocked as well because it killed a promising, fantastic football match. What for? If it's a bad tackle or a second bookable offence okay, but frankly it is embarrassing."
Barca coach Pep Guardiola was philosophical about Arsenal's viewpoint, which detracted from a wonderful performance by his team.
"I have got sympathy because I love how this team [Arsenal] play," he said. "I don't know how they will analyse it in the end - they may analyse that they lost because it was ten versus 11. It depends.
"I can understand [their view] if it is that they are out of the quarter-final because of the card for Van Persie - if they believe that then okay. I can understand."