After establishing a two goal lead against United, Burnley were pegged back 2-2 during the dying embers of the game.
And although a point at Old Trafford can be considered a positive for the Clarets, Mee has admitted that the dropped points was tough to take.
Mee said: "It's a tough one when you get so close to three points, it's a tough one to take but it was backs to the wall once we got the two goals.
"We knew they were going to come out and attack us and press us and move the ball fast. We dealt with most of it but the scruffy goal at the end was a right kick in the teeth.
"We're very organised as a unit and a team and when we're 2-0 up we feel like we can hold out and keep them out.
"We would have liked to have had the ball more but it's tough when you come away from home at a ground like this and you know you're going to be in for a game."