The Clarets could have gone up with victory over Middlesbrough at Turf Moor, where they had lost just once in the league all season, if the third-placed Rams had not defeated Huddersfield.
The Terriers took the lead against Steve McClaren's men, yet they rallied back to clinch a 3-1 win while Burnley's 1-0 loss to Middlesbrough saw the gap close to eight points with 12 left to play for.
It would take a remarkable collapse for Sean Dyche's men to slip into the play-offs from this point and Mee claims nerves will not get the better of his team-mates at this stage.
"We are actually really calm and not even thinking about it at the minute," the full-back told reporters after the defeat to Boro, which was just the second home league game Burnley have lost in 12 months.
"Because we've kept on such a level all season, we're just carrying on.
"You'd be surprised how relaxed everyone is. It showed out there; we played really well but we just couldn't score.
"It wasn't a nervy performance by any means - we went out there and we wanted to win.
"We just wanted to put on a performance and win the game, we weren't concentrating on anything else.
"Like we have done all season we just focused on the game ahead. We're disappointed not to win but the performance was there and if we play like that again we should beat teams."
Burnley now head for Blackpool on Good Friday knowing that victory at Bloomfield Road would not guarantee Barclays Premier League football.
However, it would increase the pressure on Derby, who would face Doncaster later that evening knowing anything less than three points would see Burnley go up.
"We're still in a fantastic position and we'll look towards the next game," Mee added.
"If we put in another performance like that then Blackpool are going to be on the end of a hammering.
"They are fighting for their lives and it's a local derby so it's going to be a tough game."