Solly March's goal after 15 minutes proved enough for the seventh-placed hosts on Wednesday as they moved to 42 points, the same as Liverpool but having played one game less.
Brighton have now won 12 league games this season, equalling their own top-flight record.
Palace's struggles in front of goal were evident again as they failed to take their chances with Odsonne Edouard heading a glorious opportunity wide early on.
Patrick Vieira's side are now without a goal in their last four games, without a victory in 2023 and despite being 12th in the table are only three points above the relegation zone.
At Brentford, striker Ivan Toney sent a timely reminder of his form to England coach Gareth Southgate by netting the opener in his side's 2-0 victory at struggling Southampton.
Toney reacted first in the 32nd minute to score from a Bryan Mbeumo corner and take his league tally to 16 for the season.
Bottom club Southampton dominated possession but were unable to penetrate the Bees' defence as they missed the chance to move out of the relegation zone.
Yoane Wissa sealed their fate deep into added time.
Brentford, who suffered their first defeat in 12 Premier League games at the weekend, moved into eighth place in the table with 41 points.
The Saints' best chance came just after halftime when a Che Adams effort was blocked on the line by Aaron Hickey.
The south coast club remain on 22 points from 27 games, two adrift of 17th-placed West Ham United.
Southgate will select his England squad for the Euro 2024 qualifiers against Italy and Ukraine on Thursday.
Only Norway's Erling Haaland and England captain Harry Kane have scored more Premier League goals than Toney this season.
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