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.
7. Jamie Shackleton
6 - Harried and hassled, kept it sensible in possession. Yellow card tamed him a little before he was withdrawn. Photo by Tony Marshall/Getty Images.
8. Raphinha
6 - Some inaccurate stuff gave the ball away too cheaply but always held a threat, particularly when Leeds broke. Silly challenge for the penalty. Photo by Mike Hewitt/Getty Images.
9. Jack Harrison
5 - A player not lacking in effort - he fought constantly - but struggling badly for confidence. A couple of nice moments but some very frustrating stuff on the ball. Photo by Mike Hewitt/Getty Images.
10. Tyler Roberts
7 - Did well when he got on the ball for the most part. Lovely ball in for Gelhardt's goal. Photo by George Wood/Getty Images.
11. Dan James
6 - Won a penalty, pressed well. Unable to beat men one on one when doing so would have given Leeds momentum. Photo by GLYN KIRK/AFP via Getty Images.
12. Mateusz Klich (sub)
5 - Played a part in the second goal but his late challenge on Antonio Rudiger, on a yellow, was a needless risk. Photo by George Wood/Getty Images.