Leandro Trossard missed a golden chance in the first half while Neal Maupay and Solly March were both agonisingly close to breaking the deadlock in the second half.
Fulham were also grateful to their keeper Areola for a stunning stop to keep out a sharp close range effort from Maupay after a flowing Brighton move.
Scott Parker's team then came so close to nicking all three points as Albion skipper Lewis Dunk cleared Loftus Cheek's effort off the line in the last seconds.
Scroll down and click through to see who shone and who struggled for Graham Potter's team who remain five points above Fulham.
1. Robert Sanchez 6/10
Very little to do in terms of shot stopping or gathering crosses. Distribution was sound and always looks confident
2. Ben White 7/10
Very good as always and nullified the threat of Lookman. Tidy going forward and midfield or defence...he just makes the game look easy.
3. Lewis Dunk 8/10
Brilliant last ditch clearance to earn Brighton a point. Defensively sound. A great chance with a first half header that went straight at the keeper
4. Adam Webster 6/10
Moved to the left of the defence and had one shaky moment on 35 minutes as he was caught in possession but Cavaleiro fired wide. Always looks more comfortable on the right or central