Solly March's goal inside the opening quarter of an hour has moved Roberto De Zerbi's side level on points with Liverpool, while still having a game in hand.
The result condemned Patrick Vieira's men to an 11th game without a win, as the Eagles are still yet to pick up a victory in 2023.
Neither side have moved places in the table, but Brighton are gaining ground on the sides targeting the top four, while Palace continue to get sucked in further at the bottom.
Having played a game more than many of the sides below them, Palace are just three points above the relegation zone and have now gone four games without a goal.
The win has extended Brighton's impressive record of just one defeat in their last 12, helped by the fact they have won five of their last six here at the Amex.
Eyebrows were raised just hours before kick-off when news emerged that Vicente Guaita would miss out with injury for Palace.
With Sam Johnstone also out with a calf injury, that left the visitors without a fit senior goalkeeper, so 19-year-old Joe Whitworth made his first professional start.
His opposite number Jason Steele also continued to keep Robert Sanchez out of the Brighton goal, making a fourth consecutive start.
Having been toothless in front of goal, becoming the first side since records began to not have a shot on target in three successive games, Crystal Palace looked like a completely revitalised outfit, having three inside the first 10 minutes here.
Wilfried Zaha set out his stall early on with a mazy run at the Brighton backline, forcing Steele into a stop at his near post inside 60 seconds.
The resulting corner saw the returning Cheick Doucoure hammer a shot wide, before another of the men brought into the side, Odsonne Edouard, glanced a header wide with the freedom of the six-yard box all to himself.
Michael Olise then had a chance of his own after Joachim Andersen's fabulous through ball put him through one-on-one, but Steele made himself big to deny the Frenchman.
His compatriot Edouard then tested again Steele from the edge of the box, but Palace would suffer the ultimate suckerpunch just moments later with Brighton's first real attack of note.
Some neat footwork from Kaoru Mitoma opened up the space for him to thread a ball into the path of March, who timed his run to perfection, got the ball out of his feet quickly and unleashed a wonderful strike which left Whitworth helpless in goal for the visitors.
That goal was his second in a week, taking his tally to seven in his last 11 games which is remarkable considering the wide man had netted just four in his previous 156 Premier League appearances.
Brighton took confidence from the goal and would have made it two if not for a vital last ditch interception from Doucoure, who slid in at the feet of Danny Welbeck just before the frontman would have had a simple tap in.
Both sides went close again as the half drew to a close, but the major talking point was Doucoure escaping a second booking late in the first 45, leading Vieira to take the midfielder off at the break.
Whitworth continued to justify the faith shown in him by Vieira as he kept out Pervis Estupinan's powerful drive early in the second half.
A frankly bizarre few minutes followed as referee Peter Bankes led the players off the pitch after a power cut saw most of the lighting and electronics in the stadium fail.
There was only a brief delay before the players returned, and Brighton remained on top despite the mini interruption, with Whitworth called into action once more to keep out Alexis Mac Allister's header.
The remaining half an hour went by without many clear-cut chances at either end, despite nine minutes additional time for the power cut in the second half, as Brighton saw the game out professionally with consummate ease to win 1-0.
This was the first time in seven meetings between the sides that the Seagulls had come out on top, having lost two of the last three at home despite dominating the proceedings on both occasions.
Brighton now face League Two Grimsby Town in the FA Cup on Sunday, playing for a semi-final place at Wembley which would make what has already been a fantastic season an unforgettable one for the supporters.
Palace are back in Premier League action on the same day, and face the arduous task of travelling across London to face league leaders Arsenal at the Emirates.