The Gunners were eliminated from Europe’s secondary club competition at the semi-final stage earlier this month at the hands of their old boss Unai Emery.
Since then, though, they have won four Premier League games in a row and they could still qualify for the Europa Conference League next season.
Arsenal host Brighton on the final day of the season and could climb as high as seventh if results go their way.
“After, you have to win the football matches to prove that you have reacted as a team,” Arteta said of the reaction to Villarreal.
“But the reaction, and the pain in that dressing room, it still hurts. And I think that is what we are using in the final games to show everybody, and show ourselves, that we could have done better but we are not going to give up.
“And we want to challenge to the last day to give ourselves the chance to be back in Europe, and they have the opportunity again to go.”