Bayern thrashed Atalanta 6-1 in Bergamo in the first-leg of their Champions League round of 16 clash.
Palladino told the club's website: "They are a very strong team, truly very strong. I want to congratulate them; they impressed me, they have great values. They gave us a lot of trouble, it's not the result we wanted.
"We feel sorry for the fans, who supported us from start to finish; they were the best on the pitch. We thank them for their support. Now let's put this game behind us immediately, lift our heads, recharge our batteries and focus on the championship straight away."
On the home support at the final whistle, Palladino also enthused: "It's a demonstration of great love and affection, despite the heavy defeat. They proved to be unique; they supported us until the end with a wonderful round of applause and a great chant.
"We'll try to repay them in Serie A and the Coppa Italia."