The La Liga side won 2-1 on aggregate after securing a 0-0 draw at the Emirates.
And the Gunners must now finish inside the top seven to qualify for Europe next season, with Liverpool five points ahead having played a game less and only four games remaining for Arteta’s side.
“We are devastated,” Arteta told BT Sport. “Really disappointed. We have to congratulate Villarreal. We tried everything until the last minute. I think we deserved to win the game but the details define these ties.
“We had three big chances, they didn’t have anything but they are through. So many things happened to us, and so many players were just trying to compete not at their best.”
While Freddie Ljungberg could not hide his disappointment at the defeat over two legs, urging the club to provide “investment”.
The former Arsenal player and manager told BT Sport: “I hope that this is the rock bottom and it can change from here. But it needs investment to ensure that.
“It really, really hurts but I think there needs to be a change in how they invest into the club, and change the pattern that we get worse and worse every year.”
While Martin Keown claimed a decision must be made over Arteta’s future.
“That decision [on Mikel Arteta’s future] needs to be made. He’s a young manager. It’s not a given that you will be successful even if you come with a big reputation,” Keown told BT Sport.
“You have to be a quick learner as a manager. I’m not saying he isn’t going to be manager of the club, that’s a decision that will be made through the summer.
“It will have to be an incredible start to next season if he is to stay in the job. He’s under huge pressure now.
“Is there an appetite to keep him? We will have to see what happens but these performances can’t continue. Big decisions to be made at the football club all-round at the moment.”