The two sides renewed their top-flight rivalry after 36 years in the Eagles' first visit to the Amex Stadium since their infamous victory in the Championship play-off semi-final of 2013, and the tension around the ground was reflected by two teams who defended in numbers.
The return of Christian Benteke to Roy Hodgson's starting line-up, alongside Wilfried Zaha and Andros Townsend, ensured that Brighton could not relax and all three forwards had chances to score Palace's first away goal in the league since April.
Former Palace striker Glenn Murray was given precious little to work with at the other end until six minutes from time, when he had a header cleared off the line, and Chris Hughton's side drew a blank for the first time in six league games at home.
The result marked another step along the road to recovery for Palace, after making the worst start to a season in Premier League history, and they have now taken five points from their last three league games.
Matt Ryan made a brilliant double save to deny Benteke and then Zaha after 14 minutes as the visitors exploited gaps in front of the Brighton defence, and the goalkeeper had to be alert to cut out Townsend's in-swinging cross four minutes later.
Having stemmed the flow of Palace attacks, Brighton won a corner and Lewis Dunk outjumped James Tomkins to test Wayne Hennessey with a header after 21 minutes, but the goalkeeper was equal to it.
Jose Izquierdo lost his marker on the edge of the box and stung Hennessey's palms with a dipping drive just before the half-hour mark and Palace were let off by Dale Stephens, who skewed the rebound well wide with the goalkeeper still on the ground.
Palace struggled to cope with Izquierdo's pace and he surged forward after 37 minutes before switching the play to Anthony Knockaert, who teed up Pascal Gross but the German's low drive was saved by the Hennessey diving to his left.
Brighton heaped pressure on Hennessey's goal early in the second half and Dunk pulled off a near-post back-heel that flashed narrowly wide after a corner in the 56th minute.
At the other end, as visiting supporters set off fireworks in the stand, Ruben Loftus-Cheek carved space for himself in the penalty area but his shot was headed clear by the diving Shane Duffy.
Zaha whipped a bouncing cross tantalisingly along the edge of the six-yard box, just beyond the reach of Benteke, before Brighton attacked and Duffy looped a header over the crossbar.
As Palace began to tire, Stephens tried his luck from 20 yards but his low shot flashed just wide with Hennessey at full stretch, but Loftus-Cheek still had the legs to spark a 79th-minute break that saw Townsend square the ball to Zaha, who sent his shot high and wide of the target.
With six minutes left, a Brighton corner found the head of Murray at the far post and the striker powered the ball towards goal, only to see Townsend clear the ball off the line and ensure Palace held on for a share of the points.