The Swans appeared to have taken all three points when they raced into a second half lead courtesy of Gylfi Sigurdsson and a Leroy Fer brace cancelling out Wilfried Zaha's opener.
But two sensational late comebacks first saw Palace take the lead when James Tomkins, Jack Cork's own goal and Christian Benteke gave the Eagles a remarkable 4-3 lead.
However, Fernando Llorente's late double left Palace in disbelief and boss Alan Pardew now has serious doubters over his ability to turn his side's form around.
The two sides have now conceded a combined 54 Premier League goals this season.
one-apiece in first half
Palace had the first chance of the game when Yohan Cabaye's corner was met by Scott Dann, whose header was blocked but the ball fell to Benteke who blazed over the crossbar. The Belgian could have scored with a little more composure.
The visitors took the lead when Benteke flicked a direct pass into the path of Zaha, who expertly turned past two defenders before firing low past Lukasz Fabianski.
Connor Wickham, who later left the field on a stretcher, almost doubled the lead soon after but Fabianski was equal to Palace's efforts on this occasion - diving low to hold the striker's header.
Sigurdsson then levelled the score against the run of play. Jason Puncheon conceded a foul on the edge of the Palace penalty area and, similar to this fixture last season, the Icelandic international beat Wayne Hennessey with the resulting free-kick.
The visitors almost restored their lead within minutes of conceding when Cabaye rounded Fabianski but, with the angle tight, decided to cross for Zaha whose header struck the crossbar. The rebound fell kindly, needing either Wickham or Benteke to find the net, but they rather comically got in each other's way and Swansea cleared the danger.
Modou Barrow and former Palace winger Wayne Routledge both had good chances, but each could only shoot over as the half came to a close.
Second half turns into a goal-frenzy
The first action of the second period led to Wickham being taken off on a stretcher. The striker went to ground clutching his knee and required three minutes' medical attention. Should the injury be bad, Pardew will be left with just two fit recognised strikers until the Christmas period. Andros Townsend replaced the former Sunderland man as Pardew reverted to a 4-3-3 formation.
Tomkins had to be at his very best to deny Barrow; the former West Ham defender performed a sensational last-ditched block to deny the wide man's goalbound effort.
However, Swansea - dominant in the second half - then took the lead when Fer capitalised on a vast amount of space in the area to score after Cabaye had cleared a shot off the line.
Fer then completed a quick-fire double when he tapped home from Sigurdsson's free-kick, giving his side what seemed, on paper, to be a relatively comfortable lead.
Palace's change in shape following Wickham's injury appeared to affect them but they were gifted what, at the time, appeared to be a lifeline when Tomkins scrambled over the line from a corner.
The second-half goal fest was not finished there, however, as the Eagles somehow drew level in bizarre fashion when Zaha's cross looped into the net having taken a deflection off the unfortunate Cork.
The game then took another turn as Benteke lashed home after Dann knocked down Cabaye's corner. 4-3 up, you would assume Palace would sit back and defend.
But no. Remarkably, it was then 4-4. Llorente got the faintest of touches as Sigurdsson volleyed towards goal.
One would be mistaken for thinking it all ended there. Soon after the Spaniard equalised, he then scored another; scrambling the ball into the net following a free-kick.
Thankfully, it did end there. A nine-goal thriller leaves both under-pressure managers with completely contrasting emotions. Bradley finally earns his first win, but Pardew will now face the toughest few days of his reign as Palace boss.