The game at the London Stadium was finely poised midway through the second-half, after Jarrod Bowen's flicked header had cancelled out Jean-Philippe Mateta's opener.
Step forward Mitchell, who lashed home a stunning volley for what proved to be the winner in the 68th minute.
It gave Palace all three points and pushed Oliver Glasner's side into the top four, with the wing-back the hero of the hour.
Upon watching his finish back after full-time, Mitchell told Palace TV: "I just wanted to make sure I didn't shank it!
"I got my body in the right position to get a good shot off. Because it was on my weak foot, I was probably even more focused.
"But I'm just happy it was a good connection and it went in.
"It's a beautiful feeling, that's what you play football for from when you're young."