The Viola lost 3-2 in the first-leg of their round 16 tie in Greece.
Fiorentina had fought back from 2-0 down to level at 2-2 through Lucas Beltran and Nicolò Fagioli. But Tete found a winner for Panathinaikos in the second-half.
Palladino said afterwards: "I think we had a good first half despite the approach not being what we wanted. We conceded two goals, and then there was a brilliant reaction from the team. We put the opponents in difficulty, we got back to 2-2, then there was the disallowed goal.
“And then, inexplicably, there was this second half, again with the wrong approach, which led us to conceding that third goal.
“I’m really sorry because the boys were the first ones to want to do better in the second half in the dressing room."
While disappointed, Palladino is convinced his players can overturn their deficit in Florence.
"We got a bit nervous, probably because time was running out and we got carried away by the game.
“Obviously, that shouldn’t be an excuse, it’s not an alibi, but I saw some of the guys get nervous in the second half. But, it’s not over because we have the second leg. I’ve seen the team looking disappointed but fired up, so they are eager to overturn this result.”