Danny Ings ended his barren run with the winner at Hull on the day the Clarets finally lost their fight to stay in the Premier League.
Ings’ 10th goal of the season, his first strike since finding the net at Manchester United 10 games ago, settled a nervy encounter and pushed the Tigers deep into relegation trouble.
But after relying on results from elsewhere, victories for Leicester City and Aston Villa - plus Newcastle United’s point at home to WBA – meant that even a fifth straight win at the KC Stadium could not keep Burnley’s hopes alive.
The Clarets were unchanged from the side that lost controversially at West Ham seven days earlier.
But there was a return to the bench for Sam Vokes, following a thigh injury that forced him to miss two games.
And after a confident start from the visitors, Hull slowly found their feet to apply some pressure.
City knew that Sunderland’s victory at Everton in the lunchtime game, combined with two final fixtures away at Spurs and at home to Manchester United, increased their need for three points.
And a Robbie Brady free kick that was blocked, followed by Tom Huddelstone’s deflected free kick, raised the volume levels inside the KC.
Jason Shackell’s terrific clearance inside the six-yard box, from Brady’s dangerous cut-back, then denied the Tigers a further goalscoring opportunity.
Brady then dug out a cross from the left for the best chance of the game so far, which the inrushing Ahmed Elmohamady wasted by heading tamely over at the far post.
Burnley had enjoyed plenty of possession, without cutting through the Tigers defence. And Ashley Barnes cut a frustrated figure on the half hour when he lost his marker, but headed Kieran Trippier’s inswinging corner over the angle.
And the Clarets striker’s header soon forced the first save of the game from Steve Harper, when Matt Taylor picked him out in a crowded box.
But Brady continued to be a thorn in Burnley’s side and with nine minutes of the half remaining, the left back clipped the top of the crossbar with a free kick from distance.
With wins a must for both sides, there was an element of surprise at the lack of clear-cut chances in a nervy opening half.
But things opened up after the restart - and when Barnes’ overhead kick soon bobbled inches wide gasps of relief from home fans were audible.
The Clarets striker had another chance five minutes into the half, glancing a header just wide after Taylor picked him out with a drilled left wing cross.
And within three minutes, Ings made mincemeat of Michael Dawson on the left and drove into the penalty area, where he finally fell under a clumsy challenge and appealed in vain for a penalty.
Former City star George Boyd had yet another appeal rejected after appearing to be clipped in the area.
But from the resulting corner the opener finally came.
Dawson had to leave the pitch for a shirt change and treatment to a head injury. And before the towering defender could re-join the action, Ben Mee and Boyd kept the ball alive and, as Hull failed to clear, Ings rifled the ball under Harper from eight yards.
As City went for broke, with three substitutions all made, Heaton grabbed Abel Hernandez’s audacious back heeled attempt.
There was then disbelief in the home ranks when, following Barnes’ foul on Stephen Quinn, Brady stepped up to rattle the crossbar with another delightful free kick.
But as Burnley saw out the game defending resolutely, while Newcastle ground out a vital first point in nine games, a hushed reality enveloped the home stands as City’s own Premier League survival hopes suffered a big blow.
BURNLEY: Heaton, Mee, Shackell, Duff, Trippier, Boyd, Jones, Arfield, Taylor (Ward 76), Ings (Keane 88), Barnes.
Subs: Gilks, Vokes, Kightly, Ulvestad, Jutkiewicz
Booked: Ings 79
HULL: Harper, Chester, Hudlestone, Brady, Livermore (Meyler 57), McShane (Jelavic 57), Dawson, Aluko (Hernandez 66), Elmohamady, N'Doye, Quinn.
Subs: McGregor, Rosenior, Bruce, Robertson
Ref: M Atkinson