The Whites fell behind in just the fourth minute as Mason Mount curled a fine finish into the top right corner from Chelsea's first attack.
United's task was then made even harder 20 minutes later as Dan James was shown a straight red card for a late challenge on Mateo Kovacic that eventually forced the midfielder off injured.
A dominant Blues side finally bagged a second goal ten minutes after the restart when Christian Pulisic applied a neat finish from the edge of the box and Romelu Lukaku bagged a third Chelsea goal with seven minutes left.
The result keeps Leeds in the relegation zone with just two games left.
Graham Smyth, the YEP's chief football writer, hands out the scores from another Whites defeat at Elland Road.
13. Mateusz Klich (sub)
6 - Ran around a lot, tried his hardest to get on the ball and keep it but had no options.
14. Joe Gelhardt (sub)
6 - Went in search of the ball and tried to win free-kicks in the opposition half. Produced a fine cross for Koch.
15. Anthony Taylor (referee)
4 - Got the James' red right but was incredibly lenient on the visitors all evening, which frustrated Leeds further and added needle.