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.
7. Adam Forshaw
8 - As muscular and rugged a performance as any this season. Thundered into tackles. Played a big part in the attempted fightback. Photo by PAUL ELLIS/AFP via Getty Images.
8. Mateusz Klich
6 - Lacked his shooting boots. Got into great positions, showed some tidy touches but lacked composure when it was needed. Photo by Laurence Griffiths/Getty Images.
9. Jack Harrison
5 - Missed a big chance early on. Struggled to make much of an impact after that as the visitors got on top. Photo by George Wood/Getty Images.
10. Dan James
7 - Lots of effort, a lot more influence in the second half. Set up Raphinha's goal. Tracked back well. Photo by PAUL ELLIS/AFP via Getty Images.
11. Rodrigo
7 - Somehow scored from wide on the left, played some intelligent stuff, put in lots of effort. Dropped deep to try and dictate. Photo by PAUL ELLIS/AFP via Getty Images.
12. Junior Firpo (sub)
6 - Some lovely stuff going forward and a big tackle or two. Poor touch cost Leeds for the third goal though. Photo by Shaun Botterill/Getty Images.