The Gunners were the reigning holders of the trophy but exited this year’s competition without any real impact, losing 1-0 at St. Mary’s on Saturday.
Arteta criticised his team’s start to the match, which saw Saints dominate and eventually score when Kyle Walker-Peters’ shot across goal was turned in by Gunners defender Gabriel.
“I’m really sad to be out of the competition. Congratulations to Southampton but [I’m] disappointed to be out and [in] the way we lost the game, conceding in an area where we cannot give the ball away,” the manager told BT Sport after full-time.
"We had issues in the first half controlling certain aspects and with the high press they do. We had two or three important moments but didn’t hit the target, not enough to win the game.
“The way we conceded is our own fault and you cannot do that. it’s a lesson.
“In the second half we were much more dominant and played in the opponent’s half but it wasn’t enough.”
Arsenal travel to Southamton again in midweek for the Premier League game on Tuesday evening and Arteta hopes that lessons have been learned, as they look to continue their upturn in fortunes in league play at least - though their opponents are one place above them in the table.
“Southampton demands the best out of you and the boys tried their best. I hope we can use that to be even more prepared.”
Saints now progress to the fifth round to face Wolves, who beat non-league Chorley on Friday night.