Wilfried Zaha netted either side of a James Tomkins goal to put Palace in control, and although Glenn Murray and Joze Izquierdo got on the scoresheet for the Seagulls, they could not force an equaliser in the second half.
The Eagles now sit six points clear of the relegation zone and just a point adrift of Brighton, who remain in 13th position in the standings.
Palace required just five minutes to go ahead in front of their own supporters through Zaha, who was able to tap home from a yard out after Brighton's failure to deal with a corner led to Luka Milivojevic firing a shot across goal which was only half dealt with by Mathew Ryan.
Roy Hodgson's team were in full control of the match, and a second goal arrived just before the quarter-hour mark when Tomkins forced the ball into the bottom corner from 10 yards after Brighton had initially made two blocks from goalbound efforts.
However, Brighton soon stunned Palace by halving the deficit through Murray, who was perfectly placed to lash home from inside the six-yard box after Lewis Dunk had powered a header towards goal.
Palace were soon back in the ascendancy and by the 24th minute, the Londoners had netted their third of the match as Zaha beat his marker at the back post to head into the bottom corner from Milivojevic's pinpoint cross from a central position.
Zaha tried his luck from distance shortly after the half-hour mark as he looked for his hat-trick but moments after Ryan had made a comfortable save, Brighton again found a way back into an absorbing contest.
Jurgen Locadia's clever pass on the flank released Izquierdo and after advancing into the penalty area, the Colombian curled a delightful shot into the far corner of the net.
Once Brighton had showed that they remained confident of a turnaround, the tide slowly turned in their favour as they began to have more success in the final third at a time when Palace started to pose less of a threat up the other end.
The opening stages of the second half passed by without incident but shortly after the hour mark, the Seagulls came close to equalising when Murray met a cross at the back post but his volley across goal just missed the far post.
Aside from an ambitious free kick from an acute angle from Milivojevic, Palace were struggling to work Ryan in the Brighton goal and it resulted in Christian Benteke being introduced from the substitutes' bench.
However, as the game entered the final 10 minutes, Benteke was needed to make more of an impact in defence from set pieces as Brighton began to up the tempo in the final third.
In added-on time, Murray squandered a chance from close range after the ball had been flicked into his path, and it proved to be Brighton's final opening as Palace held on for what could prove to be a crucial three points come the end of the season.
CRYSTAL PALACE (4-4-2): Hennessey; Ward, Tomkins, Sakho, Van Aanholt; McCarthur, Cabaye (Benteke 71'), Loftus-Cheek (Riedewald 80'), Milivojevic; Townsend, Zaha
BRIGHTON & HOVE ALBION (4-2-3-1): Ryan; Schelotto, Duffy, Dunk, Bong (March 86'), Kayal, Stephens; Locadia (Knockaert 46'), Gross (Ulloa 71'), Izquierdo; Murray