The Cottagers secured an early 1-0 lead as Alex Iwobi pounced on a defensive howler from Seagulls goalkeeper Bart Verbruggen, who played a pass too short in his own box.
The Albion netted an equaliser ten minutes after half time, with Joao Pedro beautifully controlling a ball over the top to pass to Baleba, who slotted home past Bernd Leno to make it 1-1.
The strike saw Baleba score his third goal in all competitions for Brighton on his 50th appearance for the club.
But an own goal from Matt O'Riley from an Andreas Pereira free-kick into the box, as well as a second from Iwobi, earned the west Londoners all three points.
Fulham have now both scored and conceded in all seven of their Premier League home games this season.
more premier league
Man United fans outraged as video of Marcus Rashford at Arsenal emerges
Gary Neville laments Man United with X-rated summary as Arsenal break club record
In their league history, only in the 1950/51 season have they had a longer such run from the start of a campaign.
Fulham also equalled their club record of scoring in 13 successive top division matches so far.
That's a feat that's stayed untouched for 65 years.
Fulham have also maintained their club-record unbeaten streak against the Seagulls in the Premier League.
Most read in Football
Major rule change to stop time wasting goalkeepers considered after successful trial
West Ham delay Julen Lopetegui decision amid approach for six-time Serie A winner
Ange Postecoglou reveals concerning reason why Dominic Solanke was sent home in new update
Cristiano Ronaldo gives savage response to Rafael van der Vaart talkSPORT interview
The west London club are now unbeaten in all nine of their Premier League games against them with five wins and four draws.
Brighton have now edged Norwich for being Fulham's side they have faced the most without losing in the Premier League.
The Seagulls have won none of their last ten Premier League away games against London clubs, drawing four and losing six.
The last time they had a run of this length without a league win in the capital was between 1961 and 1967.
Fulham are the club the Seagulls have faced the most in the Premier League without ever winning in the competition.
Going into the clash, Brighton were unbeaten when conceding first in the Premier League this season, with two wins and two draws coming their way after going behind so far.
And Fulham had won just one of their last five Premier League games in which they've scored the opening goal, drawing once and losing three.
But the Cottagers shook off those ominous omens to seal victory.
Fulham star Antonee Robinson reacts to USMNT hiring Mauricio Pochettino
Marco Silva's side have jumped from 12th place to sixth in the table.
They are now one space and one point behind Brighton, who sit in fifth.
The Seagulls have won one out of their last five league games.
Sparta Rotterdam 
Ajax 
Roma 
Parma 
Dundee Utd 
Celtic 
Valencia 
Alaves 
PSV 
Feyenoord 
Bordeaux 
Rennes 
ST Etienne 
Marseille 
Monaco 
Everton 
Chelsea 
Leicester 
Wolverhampton 
Manchester Utd 
Bournemouth 
Real Madrid 
Sevilla 
Cremonese 
Sampdoria 
Tottenham 
Liverpool 
VFL Wolfsburg 
Borussia Dortmund 
Las Palmas 
Espanyol 
Monza 
Juventus 
Lens 
Paris Saint Germain 
Fiorentina 
Udinese 
Benfica 
Estoril 
Inter 
Como 
Manchester City 
Fulham 
Nottingham Forest 
Motherwell 
ST Mirren 
Rangers 
Carrarese 
Arsenal 
Ipswich