Trailing 2-0 from the first leg in north London last month, Arsene Wengers started brightly.
However, the home side took the lead in the 18th minute when Neymar finished from close range.
January signing Mohamed Elneny smashed home an equaliser in the 51st minute but Barcelona replied 15 minutes later when Luis Suarez struck a stunning volley.
And Lionel Messi added gloss to the scoreline with a dinked finish over David Ospina two minutes from time to wrap up a 5-1 aggregate win for Barcelona and book their place in the quarter-finals.
Arsenal now turn their attentions towards the Premier League and their trip to Everton on Saturday lunchtime.
And speaking on BT Sport after the final whistle, Wright said: “I thought the boys did pretty well. It came down to chances.
“Going into Everton and the rest of the season, it’s going to come down to chances and taking chances.
“Hopefully they will learn from today. They’ve got to learn to try and take chances when they get the opportunities in games.”
Arsenal are currently third and 11 points behind leaders Leicester.
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