The Gunners were beaten 2-1 at the Emirates Stadium as the Saints fought back from a goal down to book their place in the fourth round.
While admitting that the visitors put in an impressive performance last night, Bellerin believes that Arsenal were unlucky to lose the tie.
"It wasn't our lucky night so we're out," he told the club's official website. "They played a really good game as well but I think the team played well, we had our chances.
"I think we deserved a bit more but these things happen in these kind of competitions where you've just got one game and you've got to do your best."
Arsene Wenger's side continue their Premier League campaign on Saturday with the visit of local rivals Tottenham Hotspur.