In the absence of Watford match-winner Jay Rodriguez, the skipper assumed the role of goalscorer as he converted Ashley Westwood's second half set-piece in similar fashion to his injured team-mate.
The 30-year-old, who last netted in a 2-1 defeat to Manchester City at the Etihad Stadium in January 2017, got across his marker to steer his header beyond Vicente Guaita and off the inside of the upright in the 62nd minute.
Selhurst Park hadn't always been a happy hunting ground for the Clarets. They'd won just once in 13 visits previously, when Ashley Barnes and Andre Gray scored vital goals towards the end of the 2016/17 campaign.
And they hadn't scored in nine of their last 12 games at the stadium. But the Clarets, who have now lost just once in nine Premier League outings, tore the record book to pieces as they leapfrogged their opponents in the table.
Palace's dangermen were kept quiet in a very encouraging first 45 minutes for Sean Dyche's side, who had just edged possession.
Dwight McNeil, who was vibrant and influential in a floating role off the striker, cleared the bar when Ashley Westwood's corner had caused problems inside the box.
Nick Pope collected Jordan Ayew's attempt from outside the box on the half-hour mark and Phil Bardsley got a crucial touch to Joel Ward's cross just before the break with James McArthur right underneath the ball.
The home side had more of the play after the break, but Burnley made their one and only second half opening count.
Guaita got a touch to Mee's finish, but the Spanish stopper could only turn the ball on to the inside of his post.
Andros Townsend went close when whipping the ball from right to left, with his effort just clearing the woodwork, while Luka Milivojevic header over from Wilfried Zaha's cross.
Former Claret Gary Cahill flicked a header wide late on, but the away side held on to move level on points with seventh place Tottenham.