Adam Lallana, returning from injury, had a goal ruled out by VAR and Robert Sanchez made some key saves in place of Mat Ryan.
The Seagulls remain in 16th place on 11 points while the Cottagers move out of the bottom three on goal difference.
Story of the match
Brighton manager Graham Potter made six changes from the side that lost 3-0 at Leicester City with Steven Alzate, Tariq Lamptey, Leandro Trossard and Adam Webster joining Lallana and Sanchez in the side.
The Seagulls dominated the opening 20 minutes, but there was little in the way of significant chances, Fulham goalkeeper Alphonse Areola making a comfortable save in the 15th minute.
The best sequence of the contest to date came in the 28th minute when Solly March picked out Lallana, who found Lamptey, but his pass was out of Danny Welbeck and Trossard's reach.
The Cottagers had their first real chance two minutes from halftime when Ivan Cavaleiro gathered the ball in the Brighton box and unleashed a fizzing shot that Sanchez parried away.
Brighton thought they had taken a 49th minute lead when Lallana tucked home March's cross with his left foot, only to see the goal erased when VAR ruled Welbeck had handled trying to control the ball in the build-up play.
Areola kept the game scoreless after saving Dunk's header from an inswinging corner and smothering the rebound. Another header from the Seagulls captain was cleared off the line by Tosin Adarabioyo.
Adam Webster was unlucky not to give the visitors the lead on the hour mark when his thumping header hit the underside of the crossbar and bounced out.
Ruben Loftus-Cheek tested Sanchez with a low shot that the Brighton keeper easily handled and aside from an 80th minute chance by Ademola Lookman, set up by Aleksandar Mitrovic, the match ended as it started.
Man of the match: Solly March
Quite honestly, no one deserves this "honor" in such a drab game, but Brighton were the better side and March was influential in controlling the midfield and setting up his teammates for the chances they did have.
It's the same story for Brighton: they're the better side, but for one reason or another, they can't find a way to win. Again, they created far better opportunities, but were left unsatisfied. Graham Potter will hope to see this change as soon as possible.
Fulham will be happy to escape the drop zone, but much work is to be done should they fancy staying up. Mitrovic alone won't keep them in the top flight and they lacked any real punch in the final third.