The Clarets looked to be heading to another miserable defeat and a 17th game without a win when they went 2-0 down during the first-half.
But a bonkers seven minutes before the interval turned the game on its head, as Hannibal, Jaidon Anthony and a Jefferson Lerma own goal saw them lead.
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Scott Parker’s side inevitably came under pressure during the second period, but they held firm with a spirited, backs-against-the-wall display to claim a long-overdue three points.
Here’s how Burnley’s players rated:
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13. SUB: Lyle Foster - 7/10
For Flemming, 72 Gave Burnley an important outlet in behind Palace’s defence and helped wind the clock down during the final moments. (Photo by Justin Setterfield/Getty Images) Photo: Justin Setterfield
14. SUB: James Ward-Prowse - 6/10
For Hannibal, 82 Delivered a good corner with his first touch of the ball in Burnley colours before taking a late clattering. (Photo by Stu Forster/Getty Images) Photo: Stu Forster
15. SUB: Florentino Luis - 6/10
For Ugochukwu, 87 Helped shore things up for the final stages, adding some fresh legs and making some key clearances. (Photo by George Wood/Getty Images) Photo: George Wood