It means the Premier League leaders have now been beaten in the FA Cup and the Carabao Cup final in a matter of weeks.
Arteta said: "I am very disappointed in the manner in which we lost the opportunity to go back to Wembley. We had many periods of dominance.
"We didn't capitalise on our chances enough. When you defend the box in the manner we did, you're going to put yourself in a really difficult position.
"They did capitalise on that, and that's the reason why we're out. I want to congratulate Southampton, I think they are a really good side.
"It's not a coincidence they are on the run they are. They were far more efficient than we were when it mattered. That's why they are at Wembley.
"I wish them all the best. We've played against every kind of behaviour from the opponent. We've played 51 games and won in different ways.
"I think it wasn't about that (tactics). Without taking any credit from Southampton, it was related to something that we have to do a lot better."