Palace twice trailed at Craven Cottage in an open game, only for Mitchell to score either side of half-time to twice bring them level.
In the space of less than half an hour, the wing-back doubled goal tally from last season, with the first a stunning finish from a tight angle. His second came following a goalmouth scramble after a trademark charge down the left flank.
Mitchell later went off after taking a knock while making a vital defensive clearance, but his replacement Ben Chilwell subsequently volleyed home with his first touch to win the game 3-2.
After the dust had settled on a dramatic afternoon, Mitchell revealed that forward Eddie Nketiah had played a big role in his first goal, beyond assisting him with a clever pass.
"It was a tough game," Mitchell told BBC Sport. "They were 2-1 up and we had a lot to do, but I'm happy I could contribute and get the win.
