Palmer looked back to his best in the Blues' opening league fixture, running the game as the Xabi Alonso era got underway with three points.
The 24-year-old fed in Joao Pedro to record an assist after just 31 seconds and was returned the favour in the second half as he scored through Bernd Leno's legs.
He completed five dribbles and had nine touches in the Fulham box as he recorded a 9.2 Flashscore player rating.
Next up, Chelsea host Luton in the EFL Cup second round on Wednesday, before returning to league action at home to Brighton on Sunday.
English Premier League
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