The Portugal international recently joined on loan from Atletico Madrid until the end of the season, with Chelsea paying a fee of €11m (£9.8m) to complete the deal.
The 23-year-old joined the club in time to feature in Thursday's West London derby, where Graham Potter opted to include Felix in the starting lineup.
Former Chelsea winger Willian gave Fulham a half time lead before Kalidou Koulibaly equalised in the opening stages of the second period.
In the 58th minute, Felix's Chelsea debut was cut short when he was shown a straight red card for a dangerous high tackle on Kenny Tete.
With a one-man advantage, Fulham went on to win the game 2-1 courtesy of a header from Carlos Vinicius' in the 73rd minute.
As a result of his reckless challenge, Felix created an unwanted piece of history, becoming the first Chelsea player to be sent off on their Premier League debut.
The Atletico loanee is also the first player to be given a red card on their Premier League debut since Federico Fazio was sent off for Tottenham Hotspur against Manchester City in October 2014.
Although he was given his marching orders with over half an hour remaining, Felix still had more shots on target (4) than any other player on the pitch at Craven Cottage.
Felix will now have to serve a three-match suspension, keeping him out of Premier League games against Crystal Palace, Liverpool and the return meeting with Fulham at the beginning of February.
Following the game, Chelsea boss Potter admitted that Felix's red card and the subsequent suspension is "doubly disappointing".
Speaking to BT Sport, Potter said: "It was a forward's tackle, there was no malice in it but I understand why it was a red.
"It is another blow, the hits keep coming at the moment. He was really good, you could see his quality in the game so it is doubly disappointing for us."
A third consecutive defeat only increases the pressure on the Chelsea manager, who has overseen a run of two wins in the last 10 competitive matches.
Chelsea, who sit 10th in the Premier League table, will try to return to winning ways when they face Crystal Palace at Stamford Bridge on Sunday.