The Gunners suffered a 1-0 defeat away to Southampton on Saturday, with an early own goal from Gabriel the difference between the two sides.
Although Arsenal, who made numerous changes in their starting XI, improved in the second half, they were unable to get back on terms.
“I’m really sad to be out of the competition. I’m disappointed to be out and disappointed in the way we lost the game,” he told BT Sport.
“We controlled certain aspects in the first half, but we had some unforced errors, too. We didn’t hit the target.
“The way we conceded is our own fault. We can’t do that if we want to compete in that room.
“I think it was an even game. They won a lot of second balls. They were dangerous and hold the ball up well. In the second half, we had more quality, were more dominant. We created some chances but it wasn’t enough.”
Looking forward to the Premier League encounter between the sides in midweek, which will also take part at St Mary’s, the Spaniard commented: “I hope we can be more prepared and come here and win the game.”