Winless in six top-flight games against Marco Silva's side since their promotion to the big time in 2017, the Seagulls were on their way to breaking that hoodoo through Evan Ferguson, but Joao Palhinha's stunner saw the capital outfit depart with a point.
Rather than reward the same XI starters from their first-ever European win against Ajax, Brighton boss De Zerbi unsurprisingly brought six fresh faces into the fray, while Silva made the bold decision to drop Andreas Pereira to the bench.
The alterations had little impact on Brighton's scintillating offensive play, as Carlos Baleba and Simon Adingra both tested Bernd Leno in the seventh minute, and Fulham were also a tad fortunate that the officials saw no need to investigate Joao Palhinha's 19th-minute elbow on Pascal Gross.
However, the German quickly shook off that bump to play a cute pass to Ferguson inside the D with 26 minutes played, and the Republic of Ireland youngster took one touch with his right boot before finding the bottom corner with a delicate left-footed curler.
Leno's reach was futile as Ferguson registered his 10th Premier League goal of 2023, becoming the first teenager to achieve double figures in a single calendar year since Wayne Rooney in 2005, and his strike proved the difference in a trouble-free first half for the Seagulls.
As impressive as Ferguson's strike was, Lewis Dunk almost outdid his teammate with a rocket of a free kick four minutes after the restart, but his 25-yard set-piece did not dip quite enough and struck the frame of the goal.
Attacking substitutions then arrived en masse for Fulham, but it was their midfield enforcer who got them back on level terms, as Brighton made a mess of playing out from the back and were punished for it.
Stealing possession back in the final third, Harry Wilson played a first-time pass to Palhinha inside the D, and the Portuguese powerhouse rifled a blistering strike into the top corner in the 65th minute.
The Cottagers began to enjoy their first purple patch of the match, but Brighton soon regained a measure of control and were centimetres away from regaining the lead in the 88th minute, but Antonee Robinson produced a fabulous goal-line clearance from Adam Webster's header.
Nine minutes of added time was not enough for either side to find a winner, as Brighton stayed seventh in the table, while Fulham have not moved from their 14th-placed standing either.
The Seagulls now have a trip to Everton on their calendar for November 4, while Silva's team are in EFL Cup action away to Championship high-flyers Ipswich Town on Wednesday.