Mee rose above Ethan Ampadu to head Robbie Brady's corner beyond Aaron Ramsdale, who had already made a couple of outstanding saves to keep out Ashley Barnes and Chris Wood.
Burnley defended doggedly to preserve their slender lead, though Nick Pope was only called into action when saving David McGoldrick's strike.
The home side, who have clocked up 11 points from their last five games, climbed above Brighton in the Premier League while Sheffield United were left without a win in 16 games.
The home side, who have clocked up 11 points from their last five games, climbed above Brighton in the Premier League while Sheffield United were left without a win in 16 games.
9. Ashley Westwood 6
Ran his socks off as United dominated the ball in the second half in search of an equaliser. Integral in providing protection for Burnley's back line and was often the man on the scene when the ball broke loose around the penalty area. Photo: Sebastian Frej
10. Ashley Barnes 6
His partnership with Wood is starting to show signs of getting back to how it was. Dropped off to hold the ball up and bring others into play, was a constant menace with his physicality and went close to netting in the opening minutes of the game. Photo: Jay Barratt
11. Chris Wood 6
Put in the hard yards, chasing absolutely everything in an attempt to prevent the visitors from getting the ball forward,Created the first chance of the game when flicking on Lowton's diagonal and later denied by a sensational stop from Ramsdale. Photo: Jan Kruger
12. Josh Benson 6
Still learning the game - gaining an understanding off when to keep the ball, when to give it, when to move and when to sit - but worked well enough against the ball and was generally neat and tidy in possession. Photo: Alex Pantling