It marked the Blues' first appearance in the competition since 2023, but Enzo Maresca's men were undone by a combination of misfortune and ruthless finishing from the hosts.
Trevoh Chalobah's own goal set the tone before Harry Kane struck twice to put the German champions out of reach.
Palmer, who had briefly given Chelsea hope with a smart finish in the first half, admitted afterwards that his side failed to handle key situations.
The 22-year-old's goal, coming after a neat exchange with Malo Gusto before slotting past Manuel Neuer, was his second in four days following his strike in the 2-2 Premier League draw with Brentford on Saturday.
Yet despite his form, Chelsea could not build on his effort and were punished for lapses at both ends.
Speaking to TNT Sports, Palmer said: "Yeah [frustrating]. We deserved more. We started well, had early chances but when you make mistakes at the highest level it's difficult.
"There was a lack of concentration, not managing the moments correct.
"We showed we can play against one of the best teams in the competition. We deserved more to be honest. The manager said to us before we can go toe-to-toe.
"It's a good learning curve and to be back in the competition is nice. I don't even know [what happened in the second half]. We're not coming here to be a learning curve. We showed we want to compete."