Leeds went ahead in the 28th minute when Raphinha coolly converted a penalty that was awarded after Marcos Alonso had caught Dan James with a late challenge in the box.
The Blues drew level three minutes before the interval after Alonso picked the pocket of Stuart Dallas and sent in a low cross that allowed Mason Mount to beat Illan Meslier at his near post.
Chelsea then went in front 13 minutes after the restart as Jorginho converted the game's second penalty, awarded after Raphinha was judged to have fouled Antonio Rudiger.
Leeds then looked on course to leave with a point after drawing level with seven minutes remaining, substitute Joe Gelhardt brilliantly converting a fine Tyler Roberts cross just 81 seconds after being brought on.
But United's hearts were broken in the 94th minute when a Mateusz Klich challenge on Rudiger gave the Blues a second spot kick, Jorginho then sending Meslier the wrong way for the second time.
Graham Smyth, the YEP's chief football writer, hands out the scores from an eventful afternoon at Stamford Bridge.
1. Illan Meslier
6 - A couple of first half saves aside didn't have too much to do other than face a pair of penalties and pick the ball out of the net. Pass to Dallas for the first was risky. Photo by GLYN KIRK/AFP via Getty Images.
2. Stuart Dallas
6 - Control let him down for Chelsea's first. Plenty of effort in his performance and some nice stuff going forward. Werner caused him problems early on but he settled well. Photo by GLYN KIRK/AFP via Getty Images.
3. Diego Llorente
7 - Another who defended well for the most part. Some of his forward passing was good, difficult to find targets at times. Photo by GLYN KIRK/AFP via Getty Images.
4. Luke Ayling
7 - Defended well. Rarely able to contribute offensively but there was a nice second half run. Photo by Stu Forster/Getty Images.
5. Junior Firpo
6 - Didn't always look after the ball and risked a red card but stuck at it, battled and helped push Leeds forward late on. Photo by Marc Atkins/Getty Images.
6. Adam Forshaw
8 - As committed a performance as you'll see. Charged into everything, harried Chelsea, linked the play. Photo by GLYN KIRK/AFP via Getty Images.