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.
13. Joe Gelhardt (sub)
N/A - Scored with his first touch, showed lots of confidence and strength on the ball. Unlucky not to be able to say he rescued a point. Photo by GLYN KIRK/AFP via Getty Images.
14. Charlie Cresswell (sub)
N/A - Came on to help combat Lukaku's aerial threat. Won one big header, lost another. Photo by Lewis Storey/Getty Images.
15. Chris Kavanagh (referee)
6 - Used his cards liberally, too liberally on occasion. Needed VAR help for Chelsea's second penalty. Struggled to keep a lid on the tension at the end. Photo by Marc Atkins/Getty Images.