The tie was taken into extra-time before Ez Abde struck for the visitors to earn a 2-2 draw and hand the tie to Betis 4-3 on aggregate.
Afterwards, Palladino said: "This game leaves us with a deep sense of disappointment. We really wanted to reach the final, but we were up against a team full of quality players.
"They opened the scoring with a brilliant free-kick from Antony, but we bounced back with two goals and were in control of the game in the second half.
"We even had a chance to score a third and then in extra time, in what was probably our best spell, we conceded a goal that we probably could have avoided. But the team gave their all and I have no criticism of my players.
"Luck is a part of football and I think the key moments tonight made the difference. We played a great game, but unfortunately it wasn't enough. I'm proud of my players, although in the end we lacked that spark that could have taken us further."