The Clarets were 2-0 up inside 16 minutes thanks to goals from Ben Mee and Danny Ings before in-form Palace striker Dwight Gayle pulled one back.
Jason Puncheon equalised just minutes into the second half and then Gayle struck late on for his 10th goal in all-competitions this season to seal maximum points.
The pair both scored in the come-from-behind win over Tottenham seven days ago and repeated the feat in snowy Lancashire.
Victory helped the Eagles climb three spots into 12th position while Burnley remain just a point above the drop zone.
Sean Dyche’s men have not won back-to-back home league games since April when they were in the Championship and defeat ended a three-match unbeaten spell in the league.
Tennis season seemed a long way off in snowy Lancashire, but two disputed line calls were involved in Burnley, extending their own Premier League record by naming an unchanged side for the eighth consecutive match, taking a 2-0 lead.
Kieran Trippier’s corner may have been taken from outside the quadrant, but that would not have contributed to Joe Ledley’s inability to mark Mee, who lost his man to head home at the far post from barely a yard out.
The second was more hotly contested as Joel Ward tried to shepherd Mee’s clearance into touch only for Scott Arfield to muscle the full-back out of the way, keep the ball in play – despite Palace protestations – and slide a pass inside for Ings to charge through the middle and slot past Julian Speroni.
Pardew’s decision to swap over his wingers paid off when Wilfried Zaha’s pace helped him beat two markers and, although James McArthur’s effort was blocked by the seemingly omnipresent Mee, the rebound was drilled home by Gayle.
It was Palace’s first away goal in 453 minutes, since Mile Jedinak’s effort at Swansea on November 29.
Zaha was enjoying more success the right, but on-loan Arsenal striker Yaya Sanogo, despite holding up the ball well, was given few opportunities, with his best coming from the last action of the first half with a header hooked over by Arfield.
Burnley’s inactivity for Palace’s equaliser three minutes after the interval was inexcusable, however.
Puncheon picked up the ball just inside the opposition half, but was allowed to turn and run 25 yards unchallenged and he took full advantage to beat Tom Heaton with a low shot just inside the goalkeeper’s right-hand post.
The momentum was all with the visitors now and, although Michael Keane’s volley was cleared off the line by McArthur, substitute Glenn Murray hit a post before Heaton was beaten by Gayle’s angled shot late on.