Callum Hudson-Odoi had Forest 2-1 ahead on 74 minutes after Willy Bony cancelled Chelsea's lead via Mykhaylo Mudryk. But goals from Raheem Sterling (80') and Nicolas Jackson (82') saw Chelsea win 2-3.
Palmer said: "It's a big win. We needed to win today and get the three points. When we went behind we thought 'not again'."
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On Reece James returning, he added: "Everyone knows how good he is and to have him back is a big plus. Hopefully he can stay fit and help us.
"I came to play football. I'm trying to enjoy it and keep my feet on the ground and continue to work hard.
"Getting sixth place is our goal. Hopefully we can go back to Europe."