Van Persie, declared fit for the game only earlier on Tuesday, was shown a second yellow card by experienced Swiss referee Massimo Busacca in the 56th minute when he took a shot at goal after being flagged for offside.
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His manager Arsene Wenger said he told the referee "face to face" what he thought of the decision.
"The sending-off cost us the game," Wenger, who had a furious exchange with the referee immediately after the final whistle, told reporters.
"I think two kinds of people can be unhappy those who love Arsenal and those who love football because it's very difficult to understand his attitude.
"At the moment he made his decision it was a very promising game, very interesting.
"If you have played football at a certain level you cannot understand this decision. It's impossible.
"I think Barcelona is good enough to win in a normal way against anybody. But you want to be allowed to win in a normal way here."
Van Persie said Arsenal felt betrayed.
"How can I hear his whistle with 95,000 people jumping up - how can I hear that for God's sake? Please explain that."
"We feel a little bit betrayed, when it was 1-1 it was all to play for."
Barcelona coach Pep Guardiola said he understood Arsenal's complaints.
"I have sympathy with Arsenal," he said. "They can say `we are out because we played 10 against 11' and I can understand if they believe that.
"That's one way to analyse the situation that I can understand. Our victory will be tarnished somewhat by Arsenal's complaints and in a certain way I can understand Wenger."
