Red Devils boss Ralf Rangnick's double change rescued all three points for his team after Anthony Elanga and Fred, both 66th-minute substitutes, found the net to restore the Old Trafford outfit's lead after a stunning second-half comeback by Leeds looked set to spoil the visitors' party.
Raphinha and Rodrigo both scored within a 24-second period to make it 2-2 early in the second half, after headers from Harry Maguire and Bruno Fernandes sent the visitors into the interval with a two-goal lead.
The result leaves the Whites five points clear of the drop zone after 18th-placed Watford earned three points against Aston Villa on Saturday.
Graham Smyth, the YEP's chief football writer, hands out the scores from an epic contest in the driving rain at Elland Road.
1. Illan Meslier
6 - Distribution was an issue in the first half. Stranded for the Maguire header, might think he could have done better with the third.
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2. Luke Ayling
6 - Couldn't keep the shape together defensively but worked so hard to try and play forward and kept going to the end.
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3. Diego Llorente
4 - Shaky defensively, didn't get off the ground for Maguire's goal, had a mix up with Meslier that was almost costly. Replaced at the break.
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4. Pascal Struijk
6 - Could take some blame for the fourth. Otherwise largely solid in defence. Some good bits of play at centre-half and defensive midfield.
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5. Stuart Dallas
7 - Another who put in a huge effort. Wasn't a perfect performance but it had lots of grit and some good moments.
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6. Robin Koch
6 - Some nice early touches and passes. Caught with a body-check by McTominay and although he tried to carry on, he appeared concussed and had to come off.
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