Sean Dyche's side move above Arsenal and Sheffield United and on to 45 points, level on points with Tottenham Hotspur, who have played a game less.
Despite being depleted by injuries and with another incomplete and inexperienced substitute bench, Burnley were well on top in the first half with Dwight McNeil influential in a central role.
Palace were stronger after the break but fell behind shortly after the hour mark when Mee launched himself at an Ashley Westwood free-kick and Palace keeper Vicente Guaita could only push the ball in off the post.
Roy Hodgson's team piled on the pressure in the final minutes but Burnley, who have lost just once in the last ten games, hung on to secure the three points which will give them hope of a push towards possible Europa League qualification. (Reporting by Simon Evans Editing by Christian Radnedge)