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.
1. Illan Meslier
6 - Another tough outing for the goalkeeper, who wasn't protected enough by his side against elite attackers.
2. Raphinha
5 - Got up and down the pitch as often as he could but is being played so far from where he does his best work.
3. Diego Llorente
5 - A few mistakes and shaky moments, as has been the case in a number of recent performances.
4. Robin Koch
6 - Did his utmost, tried to charge forward in search of something, anything positive.
5. Liam Cooper
6 - Was spun easily by Lukaku once, couldn't fully get to grips with him but made lots of tackles that stopped it from getting even worse.
6. Pascal Struijk
5 - A few mistakes and shaky moments, as has been the case in a number of recent performances.