The defender scored a crucial equaliser with less than 10 minutes remaining in the match after Blues midfielder Nemanja Matic was dismissed midway through the second-half
Chelsea had taken the lead inside the opening quarter of an hour thanks to Branislav Ivanovic's fourth goal in six games in all competitions, and they looked comfortable for large periods.
However, Nemanja Matic was sent off with 20 minutes remaining for reacting angrily to a challenge from Ashley Barnes - a dismissal which will likely see the midfielder miss the upcoming League Cup final against Tottenham.
Seemingly buoyed by that sending off, Burnley snatched an equaliser with nine minutes to go as Mee rose highest to head in Kieran Trippier's left-wing corner.
It was a disappointing end to the week for Chelsea, with the club having attracted negative headlines for the racist actions of a small group of supporters in Paris ahead of the UEFA Champions League clash with Paris Saint-Germain on Tuesday.
For the visitors, meanwhile, who also came from behind to claim a draw at Manchester City in late December, it represents another potentially crucial point in their fight against relegation.
Chelsea had the first sight of goal in the fifth minute when Tom Heaton tipped over Juan Cuadrado's header after a cross from Filipe Luis - called back into the side as one of four changes from their meeting with PSG in midweek.
The hosts were in front nine minutes later thanks to Ivanovic's close-range finish.
Michael Kightly was robbed by Cuadrado who fed Hazard, and the Belgian weaved his way to the byline before cutting the ball back for Ivanovic to convert from inside the six-yard box.
Ivanovic thought he should have had a penalty in the 32nd minute when his shot struck the arm of Kightly inside the area, but referee Martin Atkinson waved away the appeals.
Chelsea felt they should have been awarded a spot-kick for a second time just before the interval when Burnley captain Jason Shackell pushed Diego Costa to the floor, but once again Atkinson decided in favour of the visitors.
Thibaut Courtois was not called into serious action until five minutes into the second half, when he reacted smartly to tip Barnes' deflected shot over the crossbar.
Chelsea responded as Costa brought a save out of Heaton five minutes later, the Spain international going in search of his 18th league goal of the campaign.
Tempers threatened to boil over in the 70th minute, when Barnes went over the top of the ball in a challenge on Matic, who angrily pushed the striker to the floor and was shown a straight red card, while the Burnley man went unpunished.
Barnes, perhaps fortunate to remain on the field, thought he had found an equaliser 10 minutes later but Courtois produced a superb one-handed save low to his right.
However, from the resulting corner, Mee headed home to spark scenes of jubilation among Burnley players and fans alike.
Deep into stoppage time, Danny Ings broke away as Burnley looked to steal an unlikely winner, but the striker fired wired of the target from the edge of the area.