Luke Thomas opened the scoring for the Foxes with a beautiful left-footed volley to grab his first Premier League goal for the club. Mason Greenwood equalised five minutes later with another fine finish.
However, Caglar Soyuncu scored the winner with a powerful header and the visitors held on for all three points.
Brendan Rodgers spoke to BBC Radio Leicester after the game.
Thoughts on the win
The Foxes defended like gladiators throughout and United couldn't handle it in the end with Rodgers' side seeing out the 2-1 victory.
After what was such an important three points for Leicester, Rodgers said: "It was an extremely important win for us, on the back of the performance on Friday.
"We've got to 66 points now, so we've taken a big step. However, there's still two games to go, teams are chasing us and there's still work to do.
"We started the game very well and scored a brilliant goal."
On Thomas' goal
Thomas scored his first Premier League goal in an unforgettable way to put his side 1-0 up at Old Trafford. Rodgers was full of praise for the young left-back: "It was an unbelievable finish. He's technically very strong, got a lovely left foot. He'll never forget that.
"In our backline, we've got two really young players. People talk about United's young players but we've got a 19-year-old and a 20-year-old and both of those boys were outstanding in different ways."
Looking ahead to the FA Cup final
Now Leicester must turn their attention to the FA Cup final this coming Saturday. They have a huge chance to win their first FA Cup and the players will certainly be up for it.
Rodgers had his say: "We've done some great work to get to this stage. Now we can get back in tomorrow, off of the back of a fantastic result, to get ready to take on the Champions League finalists."
Jonny Evans has missed the last two games through injury and will be eager to get out there on Saturday: "We'll assess it over the next few days. It was just too big a risk for him this evening. But he's very determined to be out there for us this weekend."
Jamie Vardy also went off early, here's what Rodgers had to say: "He's absolutely fine. He ran himself into the ground and I wanted to close the game out. He took a blow in the first half but no, he'll be fine.
"We've got a chance for history at the weekend."