Victory had been imperative for an Arsenal side that had won only two league games since January but the abject display with Andros Townsend, Yohan Cabaye and Luka Milivojevic all scoring for Palace only heaped more pressure on manager Arsene Wenger.
Sixth-placed Arsenal, who have finished in the top four in 20 consecutive seasons under Wenger, now have 54 points from 30 games -- seven points behind fourth-placed Manchester City, who have played a game more.
"It's damaging of course," a seemingly dispirited Wenger told Sky Sports when asked about how serious the blow was to their chances of Champions League football next season. "That's the most difficult thing for us tonight."
Rejuvenated Palace, winners of four of their previous five league games under Sam Allardyce, started confidently and opened the scoring when former Tottenham winger Townsend swept the ball home at the end of a smart passing move after 17 minutes.
Arsenal's Mesut Ozil, Nacho Monreal, Olivier Giroud and Aaron Ramsey look dejected
Reuters Palace striker Christian Benteke then had a goal ruled out for offside before Cabaye's superb curled effort and Milivojevic's penalty in five second-half minutes completed a miserable night for Wenger at a raucous Selhurst Park.
Palace stay 16th but now have 34 points from 31 matches, six more than Swansea City, who occupy the final relegation place in 18th and have played a game more.
"All we can do is apologise for that performance," Arsenal's Theo Walcott said after Arsenal fans had sung 'You're not fit to wear the shirt' at their players.
"It was a great three points and a well needed-three points," Townsend said.
"Sam Allardyce has been incredible. He looks to have worked his magic again."
RESULTS:Monday, April 10
Crystal Palace 3 Andros Townsend 17, Yohan Cabaye 63, Luka Milivojevic 68pen
Arsenal 0Sunday, April 9
Everton 4 Tom Davies 1, Romelu Lukaku 23,57, Phil Jagielka 41
Leicester City 2 Islam Slimani 4, Marc Albrighton 10
Sunderland 0 Manchester United 3 Zlatan Ibrahimovic 30, Henrikh Mkhitaryan 46, Marcus Rashford 89
Saturday, April 8
Bournemouth 1 Joshua King 42
Chelsea 3 Adam Smith 17og, Eden Hazard 20, Marcos Alonso 68
Stoke City 1 Jonathan Walters 44
Liverpool 2 Philippe Coutinho 70, Roberto Firmino 72
Middlesbrough 0 Burnley 0Manchester City 3 Ahmed Al-Muhamadi 31og, Sergio Aguero 48, Fabian Delph 64
Hull City 1 Andrea Ranocchia 85
West Ham United 1 Cheikhou Kouyate 44
Swansea City 0West Bromwich Albion 0 Southampton 1 Jordy Clasie 25
Tottenham Hotspur 4 Dele Alli 33, Eric Dier 39, Son Heung-Min 44,55
Watford 0Standings P W D L F A
Pts1
Chelsea 31 24 3 4 65 25
752
Tottenham Hotspur 31 20 8 3 64 22
683
Liverpool 32 18 9 5 68 40
634
Manchester City 31 18 7 6 60 35
615
Manchester United 30 15 12 3 46 24
576
Arsenal 30 16 6 8 61 39
547
Everton 32 15 9 8 57 36
548
West Bromwich Albion 32 12 8 12 39 41
449
Southampton 30 11 7 12 37 37
4010
Watford 31 10 7 14 36 52
3711
Leicester City 31 10 6 15 39 51
3612
Burnley 32 10 6 16 32 44
3613
Stoke City 32 9 9 14 34 47
3614
West Ham United 32 10 6 16 42 57
3615
Bournemouth 32 9 8 15 45 59
3516
Crystal Palace 31 10 4 17 42 50
3417
Hull City 32 8 6 18 33 64
3018
Swansea City 32 8 4 20 37 67
2819
Middlesbrough 31 4 12 15 22 37
2420
Sunderland 31 5 5 21 24 56
20