Chris Hughton made one change from Albion's 2-1 away loss at Manchester United in the Premier League ten days ago as Mathew Ryan replaced David Button between the sticks.
Ryan was making his first start for the Seagulls since Boxing Day after returning from Asian Cup duty with Australia.
Claudio Ranieri meanwhile stuck with the same Fulham starting XI that played in their 2-1 home league defeat against Tottenham Hotspur.
Despite the game being played in a torrential downpour, Brighton were all sunshine and smiles after three minutes. Martin Montoya on the right hand side delivered a cross in for Glenn Murray who got ahead of Cottagers defender Maxime Le Marchand to clip the ball past 'keeper Sergio Rico to give the Seagulls an early lead.
This was the Brighton forward's first Premier League goal in eight games, snapping his worst run in the division for Albion, and Murray soon added his second on 17 minutes.
Fulham defender Denis Odoi could only half-clear a Solly March cross and Pascal Gross produced a fine piece of improvisation to overhead kick the ball back into the box for Murray to stab home.
Dale Stephens was then lucky not to receive his marching orders for a tackle on Andre Schurrle on 26 minutes. The Albion midfielder went in high and with his studs up on the shin of the German World Cup winning-winger, but referee Lee Probert brandished a yellow card.
Ryan was called into action for the first time in the 32nd-minute of the game. Ryan Babel's stinging 30-yard shot saw the Australian 'keeper get down low to his right to tip the ball out for a corner.
Murray then had a great chance to secure a half-time hat-trick ten minutes before the break. Gross' free-kick into the area from the right initially saw Murray misjudge his header but he recovered to get away a shot which was well saved by Rico.
The Cottagers countered this almost immediately with a chance of their own. Babel's cross found Schurrle at the far post but Shane Duffy was on hand to hack the ball off the line to preserve Albion's two-goal lead.
Three minutes before the end of the half, Davy Propper rattled the crossbar. A long ball was taken down by Murray in the area and the Seagulls target-man used his strength to hold up the ball before poking the ball to the Dutch midfielder who unleashed an exocet rocket from 25-yards out which beat Rico but found the bar.
Brighton still had time to fashion one last chance before half-time. The Seagulls were having a lot of joy down the flanks and they almost exploited this again as Gross crossed for Murray but Albion's topscorer flicked just wide.
The half-time whistle was met by a chorus of boos from the Fulham supporters, as Brighton enjoyed almost total control over their South London hosts.
The first-half performance from the hosts left a lot to be desired, but whatever Ranieri said to his charges at the break galvanised them.
Just two minutes into the second period the Cottagers reduced the deficit. A Cyrus Christie cross was headed clear by Duffy but only as far as Calum Chambers.
20-yards out from goal, the Arsenal-loanee let off a thunderous 20-yard volley that nestled sweetly into the top-left hand corner and Fulham were right back in it.
The host had their tails up and sub Lucian Vietto fired a shot just over the Brighton bar. This would prove to be a warning to the Seagulls defence but Albion did not heed it as the Cottagers found an equaliser.
After dealing with a Fulham corner, Brighton were all at sea at the back as Babel's cross into the box was misjudged or missed by seven Albion. This allowed the Cottager's topscorer and captain Aleksandar Mitrovic to head home the loose ball on 58 minutes as the Seagulls' first-half dominance completely dissipated.
Relegation-battlers Fulham thought they had found the goal to give them the lead on 65 minutes. An intricate piece of play from the Cottagers saw Mitrovic lay the ball off to Jean Michael Seri but his side-footed shot struck the base of the post with Ryan beaten.
Murray came close to sending the Seagulls ahead on 72 minutes. Montoya whipped in another cross for the Seagulls forward, but Rico produced a stunning save to deny Albion's frontman.
Brighton would be left to rue this missed opportunity as Fulham produced the most remarkable of comebacks.
Two minutes after Murray's chance, Joe Bryan found himself on the left-wing and delivered an inch-perfect cross for Mitrovic to head home his second of the night. After looking dead and buried in the first-half, the Cottagers were a team reborn.
With Fulham smelling blood, they immediately went of the charge after their third goal. Sub Tom Cairney picked up the ball and left fly with a 20-yard peach of a shot but his attempt crashed against the crossbar.
The ball bounced off the bar and into the path of Babel who played the ball across the box with a first-time pass to give Vietto a simple header with 11 minutes of the game left.
On 83 minutes Albion were suffering a severe case of deja vu as Cairney produced another stunning curled long-ranged effort that once again managed to rattle the crossbar.
Both sides exchanged chances at the death. Duffy's header at goal was well taken by Rico while at the other end Ryan produced a fine save to deny Vietto.
The full-time whistle blew and the hosts claimed a vital three points in their bid to avoid the drop. Brighton, meanwhile, are still yet to win a league game in 2019 and will be left wondering how they failed to win a game they fully controlled in the first 45 minutes.
Fulham: Rico, Christie, Odoi, Le Marchand, Ream (Cairney 26), Bryan (Sessegnon 77), Chambers, Seri, Schurrle (Vietto 45), Mitrovic, Babel. Unused: Bettinelli, Cisse, Kebano, Ayite.
Brighton & Hove Albion: Ryan, Montoya, Duffy, Dunk, Bong, Stephens, Propper, Gross (Andone 77), March, Murray, Locadia (Knockaert 71). Unused: Button, Bruno, Burn, Kayal, Bissouma.
Referee: Lee Probert