The visitors started brightly but were pegged back by Ian Holloway's side in the second half, but the hosts were unable to capitalise and suffered two injury scares
Brighton were the better side in the first half and should have gone in front early on through Jose Ulloa.
The hosts improved in the second half, but lost Glenn Murray and Jon Williams to injury, leaving Ian Holloway to wait anxiously to see if they will be fit to play in the return leg.
Gus Poyet’s side started sharpest and could have gone in front early on as Wayne Bridge beat Joel Ward and sent a low cross into the penalty area, but Ulloa could only glance wide.
The lively Williams came closest for the hosts meanwhile, sending a low effort across Kuszczak’s goal.
But Brighton had another chance just before the 20th minute as Dean Hammond rose to meet a Will Buckley cross, but headed straight at Julian Speroni, who parried the ball clear.
Hammond had another chance to give the visitors the lead from the edge of the box after David Lopez and Andrea Orlandi combined, but the midfielder flashed an effort well wide in a half otherwise bereft of genuine chances for either side.
Palace stirred after the break as Holloway made a tactical switch, shifting Wilfried Zaha to the left hand side.
And the forward almost immediately warmed the palms of Kuszczak with a tame drive, before ghosting past Inigo Calderon and flashing an inviting delivery across goal.
The hosts were well and truly on top and Danny Gabbidon nearly put them in front from a free kick, but Kuszczak tipped the defender’s header clear at full stretch.
The Eagles were dealt an injury blow during their period of dominance however, when Glenn Murray was forced off with a knee problem and replaced by Aaron Wilbraham.
At the other end, Hammond tried his luck from distance twice but Speroni was equal to both efforts and saved them comfortably.
Holloway, meanwhile, had to make another change as the impressive Williams limped off, with Yannick Bolasie coming on.
Kagisho Dikgacoi failed to capitalise on a goalmouth scramble shortly after as Brighton stood firm to more pressure from Palace.
And the Seagulls held on to ensure a win by any margin at the Amex Stadium will send them into the play-off final.