Conor Gallagher gave the Eagles a 69th-minute lead before Andersen scored into his own net three minutes from time.
Jack Butland made several fine saves to deny a Brighton side that was consistently on the front foot.
Story of the match
Brighton should have taken the lead in the opening ten minutes as Marc Cucurella intercepted Joel Ward's pass and played in Leandro Trossard, whose shot was saved by Butland.
In the 36th minute, VAR awarded a penalty to the hosts as Will Hughes fouled Veltman in the box. Pascal Gross stepped up to the spot, but Butland dove to his left to keep the match scoreless.
VAR then went against Brighton as Butland appeared to have not gathered in Adam Webster's header cleanly, but referee Rob Jones consulted the monitor and ruled Butland had both hands on the ball.
Two minutes after halftime, the Seagulls came close again as Gross swept the ball into the six-yard box and Moder met it at the near post, but his effort hit the crossbar.
Moments later, Butland again denied Brighton as Alexis Mac Allister released Moder, but the Palace goalkeeper diverted the ball over the bar.
Following more chances that came and went for the Seagulls, the Eagles took the lead as Jeffrey Schlupp kept the ball alive on the line and Gallagher volleyed into the bottom corner when it came back to him in the box.
It appeared Palace would claim all three points until Maupay sent the ball across the face of goal and Andersen accidentally diverted past Butland as he attempted to clear.
Man of the match: Marc Cucurella
Stood out in another really impressive performance. Strong with the ball and he is always looking to move forward with real purpose.