La Dea won the second-leg 4-1 to win the tie 4-3 and reach the round of 16.
A delighted Palladino stated afterwards: "An evening that will go down in history. I want to thank all my players who supported each other, as well as my staff and the club for always being there for me.
"A huge thank you also to the 23,000 fans because the stadium was fantastic today: the city deserves this.
"It was difficult to come back today. We had to be careful with their attacks in depth, and I think we played a great match, learning from the mistakes of the first leg.
"We had to hit them where they were weak, winning every duel. I saw a truly courageous team: they deserve this success."
Mario Pasalic scored and was outstanding on the night.
Palladino paid tribute, stating: "He faced a difficult season given the passing of his father, but he always responded: he is a great man.
"I have an extraordinary group of players, especially on a human level: I also want to think of (Lazar) Samardzic and (Nikola) Krstovic, who played a great game, as did the entire team. They gave me one of the best nights since I've been here."