The visitors took the lead just four minutes into the match through Yankuba Minteh, but the goal was ruled out due to the ball going out of play.
Brighton continued to push forward, with Georginio Rutter testing Fulham's goalkeeper Bernd Leno from a distance, and Mats Wieffer's header being held by Leno.
Fulham's best chance of the first half came when Harry Wilson headed the ball over the crossbar at the back post, but the match remained goalless at halftime after a disappointing opening 45 minutes.
Brighton maintained their momentum into the second half, and earned a penalty after Georginio Rutter's run into the box was cut short by Sander Berge.
The spot kick was taken by Matt O'Riley, who calmly slotted the ball into the bottom corner, scoring his fifth penalty in his last six attempts.
The goal was the 1000th goal conceded by Fulham in the Premier League.
As Fulham chased an equalizer, Brighton had opportunities to double their lead, but Yankuba Minteh and Diego Gomez both missed chances to increase their lead.
However, in the final moments of the match, Fulham scored a late equalizer through Rodrigo Muniz, who struck the ball into the bottom corner from a corner kick.
The draw denied Brighton a fourth consecutive Premier League win and left them disappointed despite dominating large parts of the game.
The match highlight was Fulham's late equalizer, which earned Rodrigo Muniz the Flashscore Man of the Match award.