The Seagulls were the better side in the first half but were unable to turn their dominance into chances and missed a great chance to take the lead when Pascal Groß saw his penalty saved. Ultimately, Brighton were punished for their wastefulness when Fulham took the lead just before half time through Andre Schurrle after a delightful assist from Jean Michael Seri.
The second half saw Fulham further their lead through the in-form Aleksandar Mitrovic, only for the Cottagers to throw their advantage away and head back to London with just the one point.
Key Talking Point
Chris Hughton has earned a reputation for being an excellent man manager and the Irishman will be keen to maintain the harmony of his Brighton squad after making numerous new signings over the summer. But is Hughton being too cautious in letting his new men off the leash?
It is understandable that Hughton wants to reward the players who have earned his trust with their strong performances last season. But the whole point of the Seagulls bothering to sign new players was surely to challenge higher up the table and there were times in this game when an injection of extra quality could have made the difference for Brighton.
The likes of Yves Bissouma and Alireza Jahanbakhsh were excellent last season and could well be the players to make the difference for Brighton, if they are only given a chance to prove themselves by starting in a match. No one can fault Hughton's loyalty to his core players, but sometimes management is about taking risks.
Starting XI: Ryan (6); Bong (6) Dunk (5) Duffy (6) Montoya (6); March (7) Propper (6) Stephens (6) Knockaert (7); Gross (5); Murray (9*).
Substitutes: Bissouma (5) Jahanbakhsh (6) Locadia (6).
Star Man - Glenn Murray
Brighton have made some high-profile signings for a club of their size during their stay in the Premier League, with Hughton keen to build a squad capable of holding its own in the top flight. So it says a lot about Glenn Murray that the 34-year-old striker remains as essential as ever to Brighton's plans.
Glenn Murray is great. One of the most average looking blokes in the Premier League and such a consistent goal scorer— Stephen Doyle (@dubsoulrebel) September 1, 2018
Glenn Murray is the greatest.— •BBC•IGBO 😈😈😈 (@AGodRizzy) September 1, 2018
Despite his advancing years, Murray was one of the hardest-working players on the pitch for Brighton, always happy to run after a long ball or hold up the play using his strength. However, Murray's target man attributes often mean his technique goes unnoticed, as shown in this game when his cheeky swivel allowed him to win Brighton a penalty.
Few would have expected Murray to become such a key player for Brighton on their Premier League journey but the striker is as indispensable as anybody else in Hughton's side right now and proved this with two more goals against Fulham. The Seagulls must do all they can to make sure he stays fit for the whole season.
Worst Player - Lewis Dunk
Lewis Dunk was a rock at the back for Brighton alongside Shane Duffy last season but the defender had a nightmare on his return from injury as he gifted Fulham their second goal with a horrendous error.
Mitrovic just dunked on Lewis Dunk— Kyle Bonn (@the_bonnfire) September 1, 2018
Dunk found himself bullied throughout by the intimidating presence of Aleksandar Mitrovic and it ultimately proved costly when he was pressured into giving the ball away for the Serb to race through and score. A baptism of fire for Dunk on his Premier League return.
Key Talking Point
Fulham did excellently to find themselves 2-0 up after doing their fair share of defending against Brighton but once again it was their defensive vulnerability that cost them as they allowed their opponents back into the game.
Slavisa Jokanovic has built a side more than capable of performing at this level and some of Fulham's creative play has been excellent at times in these early matches. But teams like Brighton are the sort of teams that Fulham should be viewing as targets and the Cottagers need to tighten up at the back if they are to get that vital number of points this season.
Starting XI: Bettinelli (7); Le Marchand (5) Mawson (6) Odoi (6) Fosu-Mensah (6); Seri (7) Mcdonald (6) Anguissa (6); Schurrle (6) Mitrovic (7) Vietto (6).
Substitutes: Sessegnon (6) Johansen (6) Chambers (N/A)
Star Man - Jean Michael Seri
A lot of media coverage will be given to Fulham's £100m summer spending spree as the season goes on, especially if the Cottagers struggle in the league. But after four games, it's clear that Fulham have uncovered a gem in signing Seri.
The midfielder was once again at the heart of every Fulham attack during the game, causing Brighton a lot of problems with his passing and setting up Schurrle to open the scoring with a delightful chipped through ball just before half time.
Like the look of Jean Michaël Seri. Looks a terrific player.— Jigs (@Jignashm) September 1, 2018
Jean-Michael Seri is a top class player. His goal last week, and a delicious chip assist today. Just a brilliant addition by Fulham— Kyle Bonn (@the_bonnfire) September 1, 2018
Fulham have an abundance of good players but Seri has the creativity to keep their attack well supplied on his own and the midfielder could be set for a big season if he keeps up this form.
Worst Player - Maxime Le Marchand
Another new Fulham signing earned some attention for all the wrong reasons today: namely, defender Maxime Le Marchand.
The full-back was simply not at the races in this game and looked defensively suspect throughout, finally being punished for his sloppiness when his error allowed Anthony Knockaert to break through and set up Glenn Murray to pull one back for Brighton.
Fulham return after the international break with a tricky away match against Manchester City, meaning it will be vital that Jokanovic works on his side's defensive play with instant results.
Meanwhile, Brighton will be looking ahead to the short trip along the south coast to face Southampton, who beat Crystal Palace 2-0 to earn their first victory of thew new season.