Wilfried Zaha put the Eagles ahead after 28 minutes with a cool finish from 20 yards but the Bluebirds were given a lifeline when Martin Kelly put through his own net three minutes later.
They were unable to take advantage of that reprieve, however, falling behind to Michy Batshuayi's strike after 39 minutes.
Despite an improved second-half performance, which saw them go close through Josh Murphy and Kenneth Zohore, their fate was sealed when Andros Townsend added a third 20 minutes from time.
Bobby Reid netted in the last minute but the damage had been done as Cardiff turn their attention to another season back in the Championship.
Cardiff started in blistering fashion, striking the post inside 40 seconds through Murphy's deflected strike from 25 yards.
Palace eased their way into the game, however, and forged ahead courtesy of Zaha's sixth goal in his last seven away games. The Ivory Coast international collected Townsend's pass on the edge of the penalty area, created a yard of space and drilled low past Neil Etheridge.
The hosts pulled level just three minutes later, though, when Kelly turned Leandro Bacuna's right-wing cross past Vicente Guaita, who had looked well placed to safely scoop up the ball.
Parity did not last long, Palace restoring their lead six minutes before the interval when Batshuayi lashed into the roof of the net after a flowing move had carved Cardiff's defence open.
Murphy and Zohore were punished for squandering glorious openings in the 70th minute when Townsend picked the ball up on the halfway line, cut in from the right wing and drilled a low strike in off Etheridge's left-hand post.
The Bluebirds had opportunities to mount a dramatic comeback, Nathaniel Mendez-Laing hitting the post and Junior Hoilett blazing over, before Reid eventually whipped in a fine strike from 20 yards - but it did little to cheer the despondent home fans.
What does it mean? Bluebirds fail to outrun woeful start
The difficulty of Cardiff's task was made clear during an eight-game winless run to start the campaign. They found their feet after that chastening baptism but it has proved too little, too late and an immediate return to the Championship awaits. Whether Warnock, who received generous applause from the home fans after the full-time whistle, is in charge remains to be seen.
Palace, meanwhile, make it three away wins on the spin and can start planning for what they hope will be a more consistent campaign next time out.
Zaha hits new heights
The 26-year-old was at his mesmerising best, causing problems for Cardiff's defenders throughout. His strike takes him to 10 in the Premier League this season – his highest ever return in the competition.
Kelly's howler does not prove costly
The former Liverpool defender was starting just his 11th Premier League game of the campaign and it showed with his shocking own goal. He will have breathed a huge sigh of relief when Batshuayi hit the net soon after.
Key Opta Facts:
- Cardiff have been relegated, becoming the only team to play in more than one Premier League season and be relegated in every single one (also relegated in 2013-14).
- Palace have won 29 away Premier League points this season (W9 D2 L8) – only Manchester City, Liverpool and Tottenham have won more.
- Warnock has suffered his third top-flight relegation, going down with three different teams in three different decades – Notts County in 1991-92, Sheffield United in 2006-07 and Cardiff in 2018-19.
- Only Jamie Vardy (10) and Harry Kane (11) have scored more away Premier League goals this season than Zaha (nine).
- Kelly has scored each of Palace's last two own-goals in the Premier League, scoring one today and against Chelsea in March 2018.
Warnock faces the difficult task of lifting his players for the visit to United next weekend, while Hodgson's side finish their campaign with the visit of Bournemouth.