The Blues took a 1-0 lead and looked to be on course for their 13th home victory of the season until Nemanja Matic was shown a straight red card in the 70th minute, and Burnley capitalised with 10 minutes remaining.
Chelsea began brightly, though, and Juan Cuadrado forced Tom Heaton to tip his header over the bar after five minutes, while Ashley Barnes thumped a volley down the throat of Thibaut Courtois in a lively opening.
Chelsea went ahead after 14 minutes when Eden Hazard drove past four Burnley challenges inside the box and cut the ball back for Branislav Ivanovic, who slotted the ball inside the near post from six yards out.
Hazard's quality almost forced a second goal a minute later as he teed up Oscar, who was denied by a sprawling Heaton save.
After half an hour, Ivanovic sent another shot towards goal which seemed to strike the outstretched hand of Michael Kightly, but the referee waved away Chelsea's claims for a penalty.
Barnes almost snatched a goal back in the 50th minute, but Courtois reacted well to tip the forward's deflected strike over the bar.
However, Chelsea continued to forge plenty of their own chances, with Ivanovic heading narrowly wide before Costa drew a diving save from Heaton.
A few moments later, Costa came within inches of connecting with Ivanovic's curling cross from the right wing, but the ball rolled harmlessly to safety as a second goal continued to escape the Blues.
Burnley continued to dig in, though, and were handed a lifeline with 20 minutes remaining as Matic was shown a straight red card for his aggressive reaction to a raking challenge by Barnes.
Ramries then lashed a powerful effort wide before Courtois produced a fine save to push Barnes's low strike behind, but Mee nodded home at the far post from the resulting corner.
With the last kick of the game, Burnely could have won the match as Danny Ings raced away from the Chelsea defence and cut inside on his left foot, but he sliced his effort high and wide.