More than 20,000 supporters packed out Selhurst Park as Dougie Freedman's side recorded their sixth win in seven games on the club's Family Day.
Freedman made three changes to the side that overcame Wolves on Tuesday, with Andre Moritz, Peter Ramage and Kagisho Dikgacoi returning to the starting line-up, but saw his side fall two goals behind early on before a Wilfried Zaha inspired turnaround meant a 4-3 victory for the Eagles.
Zaha began the game well testing David Edgar down the Burnley left and with Darcy Blake overlapping whenever possible the home side posed a real threat. Yannick Bolasie, Moritz and Zaha all tried their luck from range early on but failed to properly test Lee Grant and the game took a while to come to life, despite Palace's early dominance.
As in the previous home game against Cardiff though, Freedman's side were soon behind as Blake's slip at the back post allowed Junior Stanislas time to curl a shot goalwards that Julian Speroni could only tip onto the post. Chris McCann reacted quickest to the rebound and poked him to give the Clarets the lead. Then, as deja-vu, hit the Selhurst regulars, the visitors took a 2-0 lead.
Just as against Cardiff less than three minutes passed between Palace conceding again, this time Martin Paterson the scorer as he took advantage of good work from Marvin Bartley to tuck a fantastic volley past Speroni and double the visitors' lead.
Despite the deficit though Palace poured forward and got their reward five minutes from the break when Moritz led a counter-attack with a wonderfully-weighted through-ball for Zaha. The young winger held off Kieran Trippier with ease and slotted his finish round Grant to reduce the deficit.
Dean Marney and Charlie Austin combined well for the Clarets as they continued to show notice of their threat but a good block by Dikgacoi allowed Speroni clear comfortably as Palace's went in at half-time with the wind in their sails.
The second period was only eight minutes old when Palace's endeavours were rewarded. Moritz was the creator again, finding Zaha on the break with another well weighted pass and the hero of Tuesday's win over Wolves made no mistake, tucking the ball between Grant's legs to level the scores.
Damien Delaney then showed why he has quickly become a fans' favourite at Selhurst, first throwing himself in front of Charlie Austin's strike and then at the other end burying a left-footed drive into the top corner after a Palace set-piece to complete an amazing turnaround.
That wasn't that though with Glenn Murray turning home Dean Moxey's cross to send the locals into dreamland with the fourth after the full-back had linked well with Jonathan Parr as Freedman's side turned the screw. Burnley kept the home faithful on their edge of their seats with the league's top scorer Charlie Austin tucking away a late third for the visitors, but that wasn't to be enough for Eddie Howe's side as Palace held on to move into fourth place after clocking up their fourth win in a row.
Crystal Palace: Julian Speroni, Yannick Bolasie (Jonathan Williams 64) (Jonathan Parr 74), Kagisho Dikgacoi, Darcy Blake, Mile Jedinak (c), Wilfried Zaha, Glenn Murray, Dean Moxey Damien Delaney, Andre Moritz (Owen Garvan 83), Peter Ramage
Subs Not Use: Lewis Price, Joel Ward, David Goodwillie, Aaron Wilbraham
Burnley: Lee Grant, Kieran Trippier, Jason Shackell (c), Ben Mee, David Edgar, Chris McCann (Cameron Stewart, 71), Dean Marney, Martin Paterson (Sam Vokes 71), Junior Stanislas (Ross Wallace 84), Charlie Austin, Marvin Bartley
Subs Not Used : Brian Jensen, Kevin Long, Brian Stock, Joseph Mills
Bookings: Trippier, Edgar