Potter's troubled side crashed to a seventh defeat in their last 10 games in all competitions after another spluttering display.
Chelsea fell behind to a goal from their former winger Willian in the first half at Craven Cottage on Thursday.
Kalidou Koulibaly equalised after the break but Felix was dismissed for a reckless studs-up lunge on Kenny Tete.
Fulham were gifted their late winner by Chelsea keeper Kepa, who made a hash of dealing with a cross that Carlos Vinicius converted to leave Potter facing renewed talk of the sack just four months after he replaced the axed Thomas Tuchel.
Potter this week labelled managing Chelsea as "the hardest job in football" after a turbulent period that is severely testing owner Todd Boehly's hope that the former Brighton boss is the right man for the task of leading the new era at Stamford Bridge
Chelsea are languishing in 10th place – 10 points adrift of the top four – after their wretched run extended to one win in their last nine league games.
They have managed just four goals in their last eight games in all competitions and Potter desperately needs a victory against Crystal Palace on Sunday to buy himself at least a little breathing space.
Hampered by a debilitating injury list that includes N'Golo Kante, Christian Pulisic, Raheem Sterling, Ben Chilwell and Reece James, Potter admitted this week that Chelsea have been "suffering".
The 47-year-old held a meeting with senior players Cesar Azpilicueta, Thiago Silva, Jorginho and Mateo Kovacic on Tuesday in an attempt to calm the crisis.
But Chelsea fans' patience is already wearing thin and losing to sixth placed neighbours Fulham was another bitter blow to his hopes of avoiding the sack.
It was in keeping with Potter's misfortune that Felix's promising debut ended prematurely after his loan move from Atletico Madrid on Wednesday.
The 23-year-old, who cost Chelsea a reported €11 million loan fee, was one of four changes from the 4-0 FA Cup defeat at Manchester City last weekend