The centre back netted in the 62nd minute when converting Ashley Westwood's set-piece to seal a fifth win on the road for the Clarets on his 300th league appearance for the club.
The 30-year-old then contributed to a 13th clean sheet of the campaign as goalkeeper Nick Pope regained the lead in the race for the Golden Glove.
Here's how we scored the players on Monday...
1. Nick Pope 7
Made saves from Ayew, van Aanholt and Cahill and aided his defence with some key interventions when Palace peppered the Burnley box with crosses and set-pieces.
2. Phil Bardsley 6
A typical blood and thunder display from the full back. Needed to be alive when faced with the threat of van Aanholt and Zaha and stuck to his task well. Threw himself into everything and made a key intervention to deny McArthur late in the first half.
3. James Tarkowski 8
Dominant in the air, nicked the ball time and time again when it was fed into his opponent and cut out everything Palace played into the penalty area. Booked in the second half when conceding a free kick right on the edge of the box.
4. Ben Mee 8
Not all heroes wear capes. The Burnley captain was outstanding at the back, putting his body on the line to keep the home side out, as well as heading and kicking everything that came his way. Took his goal superbly and he was the deserved match-winner.