The Gunners fell at the semi-final stage thanks to a goalless draw at the Emirates on Thursday, with Villarreal advancing after a 2-1 win in the first leg.
Keown's former Arsenal team-mate Dixon told BBC Radio 5 Live: "As an Arsenal fan you hope the season would change tonight and they'd put a performance in they can be proud of, but in the back of your mind you think about performances that have gone before and you think, 'I don't trust this team'.
"You think well they can't play as bad as they did in the first leg, well they've outdone themselves, because they did. They've gone out with a whimper tonight.
"Every one of those players should look at their performance and ask themselves whether they did enough, and if they're honest with themselves the answer will be no. I hope they're in that changing room hurting, but I'm not sure if they are."