Burnley were relegated from the Premier League on Saturday, despite a 1-0 victory at fellow strugglers Hull City.
Sean Dyche's men headed to the KC Stadium knowing only a win would be enough to keep their slim survival hopes alive and Danny Ings' first goal in 11 matches ensured they kept their end of the bargain.
However, wins elsewhere for Leicester City and Sunderland, as well as a draw for Newcastle United, consigned Burnley to the Championship after one season in the top flight.
Robbie Brady hit the crossbar as Hull looked most likely to find the net in the first half, but Burnley's need for three points began to tell during a frantic second half.
The visitors eventually went ahead through Ings after 62 minutes, before Brady found the woodwork again with another free-kick.
Hull were unable to find an equaliser, though, and sit two points adrift of safety with two matches left to play.
Despite losing their respective fixtures last time out, both managers opted to name an unchanged XI, with Michael Duff included for Burnley after having last weekend's red card for a challenge on West Ham's Cheikhou Kouyate rescinded.
Hull looked the more dangerous in an open first half and Jason Shackell was in the right place at the right time to clear a Brady cut-back in the six-yard box after 20 minutes.
Ahmed Elmohamady headed narrowly over from a Brady cross as the home side continued to threaten, before Ashley Barnes cleared the crossbar at the other end from a rare Burnley chance.
The hosts went close again via the head of Paul McShane, while Barnes drew a save from home goalkeeper Steve Harper with Burnley's first effort on target.
But it was Brady who came closest to breaking the deadlock before half-time, curling a 25-yard free-kick onto the bar.
Both sides went on the attack from the off in the second half – Barnes sending an overhead kick wide before Tom Huddlestone shot straight at Tom Heaton.
Barnes looked the most likely source of a goal for Burnley and would have seen his side in front had he made any kind of contact with an excellent Matt Taylor cross five minutes after the break.
Ings was left screaming for a penalty when he surged into the penalty area from the left only to go to ground in the centre of the box under the challenge of McShane, but referee Martin Atkinson waved away his protests.
Hull boss Steve Bruce threw on Nikica Jelavic – back from a knee injury – in a bid to push for the win, but saw Ings fire Burnley in front just after the hour mark as the hosts failed to clear from a corner.
Michael Dawson had been ordered from the field for a change of shirt and Ings pounced on the man advantage to drive home from the centre of the box.
Hull pressed for a leveller - substitute Abel Hernandez having a backheel saved by Heaton before Brady found the crossbar from another free-kick - while Burnley's Barnes had a goal chalked off for bundling over Harper in the process.
But the result was ultimately bittersweet for Burnley, who go down despite a third ever away win in the Premier League.
