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.
13. Raphinha (sub)
7 - Helped create chaos in the second half. In the right place to score a goal. Put in a big effort to try and rescue the game. Photo by Laurence Griffiths/Getty Images.
14. Joe Gelhardt (sub)
7 - Gave Manchester United problems when he got on the ball. Showed plenty of confidence, took on defenders and shots from distance. Photo by Laurence Griffiths/Getty Images.
15. Paul Tierney (referee)
4 - The game got away from him, couldn't retain a grip on it. Missed things for both sides. McTominay could have had three, maybe four yellow cards. Photo by Shaun Botterill/Getty Images.