Another Benteke goal and an instinctive finish from Marc Guehi swung the game back into Palace's favour, just before half-time, and then a moment of magic from Cornet levelled the scores in the second-half.
Story of the match
Crystal Palace started the match strongly and it wasn't surprising that they found themselves ahead.
Connor Gallagher's ball to the back-post was only half-cleared by Maxwel Cornet. Joachim Andersen fed the loose ball to Christian Benteke and the Belgian prodded a finish through a rack of bodies and into the net, via the inside of the post.
It was an avoidable goal from a Burnley perspective but they switched off again just moments later, allowing Gallagher in on goal, but Pope was on-hand to make a fairly routine save. It was a big chance for the in-form midfielder but his finish was rather whimsical and straight at the goalkeeper.
The signs were starting to look ominous for Burnley, who were struggling against a possession-orientated Palace team, but it didn't take long before they found themselves back in the game.
The equaliser sparked Burnley into life and Johann Berg Gudmundsson forced Guaita into a save at his near post with an audacious back-heeled effort.
Momentum had shifted and, on 27 minutes, the Clarets completed a remarkable mini-turnaround.
Dwight McNeil's floated free-kick was headed across the box by James Tarkowski and Chris Wood was positioned well to divert a header away from the dive of Guaita and into the corner. A crazy turnaround!
Burnley nearly bungled their lead less than a minute after nudging ahead, but Benteke headed wide from a glorious position after an excellent whipping cross from Gallagher.
That same combination did inevitably make Burnley pay, though, with Gallagher once again pulling the strings.
The on-loan Chelsea man dragged Charlie Taylor out of position before spinning him inside-out and slipping Benteke free. The striker made no mistake this time – smashing the ball past a helpless Pope to grab his second goal of the game.
The defending from Burnley left a lot to be desired but their defensive organisation plummeted to a new low shortly just before half-time as Palace took the lead.
A well-worked corner resulted in Cheikhou Kouyate knocking the ball for Andersen, who saw his free-header palmed by Pope. The Burnley defence stayed rooted on the line and simply stood and watched as Guehi slammed home the rebound.
Burnley came out of the traps strongly and Gudmundsson arrowed a half-volley narrowly wide of the near-post.
Gudmundsson's delivery was flicked across by Tarkowski and Cornet almost took the net off with a sensational volley from a tight angle, dragging Burnley back into this tug-of-war contest. It was a goal reminiscent of the Robbie Blake winner against Sir Alex Ferguson's Manchester United team in the 2009-10 campaign. Sublime!
The goal roused the home fans and that backing seemed to spur the players forward.
Gudmundsson tested Guaita from the edge of the box and Brownhill's fizzing cross also stung the Palace goalkeeper's gloves.
Vieira reacted by replacing the ineffective Ayew with Michael Olise in a bid to halt the momentum shift and the Eagles did eventually weather the storm.
The home fans were enjoying to jeer Wilfried Zaha, who had been involved in some little scuffles with Tarkowski and Taylor, but the winger nearly made them eat their jeers with a ferocious 20-yard volley that Pope managed to flick onto the crossbar.
At the other end, Wood forced Guaita into a routine diving save before heading over an inviting free-kick from McNeil.
Vydra nearly made another decisive contribution in the final seconds, finding himself in on Guaita after brilliantly controlling a looped ball up to him, but his shot lacked power and the keeper turned it around the post.
In the end, neither team could manage to make a decisive killer blow and both had to settle for share of the spoils.