Arsenal were held to a goalless draw by Villarreal at the Emirates on Thursday, with former boss Unai Emery's current side advancing to a final against Manchester United 2-1 on aggregate.
The exit leaves the Gunners, ninth in the Premier League, on the cusp of facing a first season without European football in 25 years -- and their fifth consecutive campaign outside the Champions League -- as pressure grows on Arteta.
"What I felt last night was a deep pain," said the Spaniard, who won the FA Cup last season in his first campaign in charge of the club.
"The moment the club is in at the moment, with everything it has been through over the last months, I had a feeling that if we could take the team to the final, it could be a great turning point.