The Clarets had the better of the first-half chances, with a clearance from Palace centre-back Gary Cahill nearly cannoning in off team-mate Patrick van Aanholt before Dwight McNeil twice went close.
Mee broke the deadlock after 62 minutes, stooping low to head Ashley Westwood's free-kick as Vicente Guaita could only palm it in off a post.
Jordan Ayew headed over from close range with 11 minutes remaining as Palace were unable to stop Nick Pope from keeping his 13th clean sheet of the Premier League campaign and moving clear in the Golden Glove race.
Palace drop to 11th position, with 42 points.
Crystal Palace: 4 Jul v Leicester (A)
Burnley: 5 Jul v Sheff Utd (H)