Jeffrey Schlupp put Palace ahead in the 22nd minute and Tyrick Mitchell's second Palace goal sealed it in stoppage time as Burnley suffered a club-record eighth consecutive top-flight home defeat, a run that dates back to the final two fixtures of the 2021-22 campaign in which they were relegated.
A similar fate will surely await if Vincent Kompany cannot come up with answers quickly as his side again could not find any cutting edge - now without a goal in six of their 11 Premier League games.
It was not for lack of trying - Burnley had 67 per cent possession and 16 shots to Palace's four - but it was the visiting fans singing loudly at the final whistle as Palace recovered from back-to-back defeats to move into the top half of the table.
Turf Moor was an almost impenetrable fortress during last season's promotion charge as Kompany's side won 16 and lost only one of 23 home games and, even when suffering relegation in 2021-22, Burnley lost only eight home league games all season. So far this season there have been no home comforts at all.
Burnley started brightly, with Vitinho winning a corner inside the opening 20 seconds before Zeki Amdouni headed a Johann Berg Gudmundsson ball wide, then narrowly missed the head of Jay Rodriguez - making his first league start of the season - with a cross of his own.
The Clarets were playing some lovely flowing football but without the end product they so badly need and they were punished as one of the best halves of football Burnley have produced this season nevertheless ended with Kompany's side trailing.
Palace had barely been seen as an attacking force - the most dangerous they had been was when Odsonne Edouard clattered into James Trafford as the Burnley goalkeeper tried to play the ball inside his own goal area, earning a yellow card.
But they got a huge helping hand when Jordan Beyer, back in the Burnley defence after a month out with injury, got himself into a tangle as he tried to play the ball back to Trafford, allowing Jordan Ayew to nick the ball away and play in a low cross for Schlupp to stab home.
It was only the second first-half goal Palace have scored all season, but the 12th conceded by Burnley, the most in the league.
Burnley tried to muster a response. Luca Koleosho controlled Charlie Taylor's chipped pass with a lovely touch, but Andersen slid in just in time to prevent a shot, and in first-half stoppage-time Koleosho saw a shot palmed over as they tried to catch Sam Johnstone out with a quick free-kick.
Standing in the east Lancashire rain, Kompany tried to inject some energy into his players early in the second half.
Rodriguez headed a deep cross from Dara O'Shea wide but Palace were defending in numbers and leaving no path through.
Roy Hodgson sent on Eberechi Eze, making his return after five weeks out with a hamstring injury, but the hosts stayed on top and should have levelled moments later when Koleosho again burst down the left and pulled the ball back only for Gudmundsson to turn it wide.
Rodriguez had the ball in the net in the 64th minute but did not need the assistant's flag to tell him he had been offside when Amdouni's blocked shot fell for him.
Still Burnley kept pushing. Koleosho blazed over before Johnstone tipped over Brownhill's dipping shot, then saving Rodriguez's header from the resulting corner.
But it was Palace who had the final say. O'Shea failed to clear a long ball in the corner and Eze laid the ball off for Mitchell to rub salt in Burnley wounds.