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.
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.