The forward scored yet another late winner to earn Alan Pardew's side all three points at Turf Moor, despite the Clarets racing into a 2-0 lead early in the first-half
The pair both scored when Palace came from behind to defeat Tottenham 2-1 last weekend, and Gayle's brace either side of Puncheon's strike saw Alan Pardew's side mount a stirring comeback once again.
Ben Mee's header - only the defender's second senior goal - put Burnley ahead inside the opening quarter of an hour, and then Danny Ings struck his seventh goal of the campaign after being set up by Scott Arfield.
The nature of that second goal left Pardew aggrieved as he felt Arfield had fouled Joel Ward in the build-up, but referee Phil Dowd saw nothing wrong as Burnley took a commanding lead.
Palace improved from that point on, however, and pulled one back just before the half-hour through Gayle's powerful shot.
Early in the second half, Puncheon, who netted the winner against Spurs, struck from distance to equalise before setting up Gayle three minutes from time for the striker to net a dramatic winner and cap a remarkable turnaround.
It was the second time in four days that Burnley had let a two-goal lead slip, after losing 4-2 to Spurs in the FA Cup third-round replay in midweek - and the defeat leaves Sean Dyche's side a point above the bottom three, while Palace climb to 12th.
Dyche named the same starting 11 for the eighth league match in succession while for Palace there was a debut for on-loan Arsenal striker Yaya Sanogo.
Following a quiet start to proceedings, Burnley took the lead in the 12th minute.
Kieran Trippier swung over a corner from the left and as Julian Speroni got caught under the ball, Mee was left free to nod home from an acute angle at the far post.
And the hosts' advantage was doubled just four minutes later as Ings finished coolly following persistent work from Arfield.
The midfielder chased a long clearance by Tripper and robbed Ward on the left touchline, before playing in Ings who shot low beyond Speroni into the right-hand corner.
After that quickfire double blow, Palace regained a measure of composure and halved the deficit in the 28th minute when Gayle rifled home after the hosts failed to deal with Wilfried Zaha's dangerous cross from the right.
The visitors continued to go in search of an equaliser as the half drew to a close, James McArthur bringing a save out of Tom Heaton before Sanogo's header was hacked away by Arfield.
Palace picked up where they left off at the start of the second half, and drew level within three minutes as Puncheon picked up the ball from Gayle and drove into space before firing into the bottom left-hand corner from 25 yards.
As the second half wore on, Burnley looked the more likely to find a winner and they went close 12 minutes from time when Michael Keane's goalbound shot was cleared from just in front of the line by McArthur.
Substitute Glenn Murray then struck the post for Palace, before there was one final twist as Puncheon's ball was allowed to travel to the back post to Gayle, who took a touch and smashed the ball low and hard beyond Heaton to spark scenes of jubilation among the visiting players and supporters.