Chelsea had a first-half lead through Joao Pedro, but La Dea fought back after the break and struck through Gianluca Scamacca and Charles de Ketelaere.
Palladino said afterwards: "We’re really happy, it’s been a magical night, shared with our fans. It was truly an unforgettable evening. We’re especially proud because we faced a very strong team, full of great individual players.
"We’re also happy because we’ve regained the DNA we had lost last Sunday in the league. I saw the team I wanted to see. We played well technically, showed great quality, and really challenged Chelsea.
"Of course, we’re happy because this is a win we wanted, a performance we were aiming for, and together as a team we gave everything to achieve it.
"Despite this important win, we must never let our focus drop in the next two games, because lapses in concentration can lead to poor performances in the league. In the Champions League we get excited and can even push beyond our limits, but the attitude and spirit we showed tonight needs to be there in every game. I want my team to always give their absolute best, just like we did tonight."