The result also means Manchester City have clinched the 2020-21 Premier League title, with United on 70 points from 35 matches, 10 behind Pep Guardiola's side, with three matches remaining
With a rescheduled game against Liverpool on Thursday, Ole Gunnar Solskjaer fielded a heavily rotated squad, with an in injured Harry Maguire not in the XI and facing a fitness race to be ready for United's Europa League final against Villarreal at the end of May.
Leicester opened the scoring when Youri Tielemans poured forward into space on the right flank, swinging in a teasing cross to the back post, where Luke Thomas side-footed a first-time left-foot volley into the far top corner for an early lead.
However, Man United were level five minutes later after Mason Greenwood picked up a pass from Amad Diallo and slalomed his way through the Leicester defence and into the area to tuck his well-placed shot just inside Kasper Schmeichel's far post.
Çaglar Soyuncu's bullet header from a corner kick shortly after the hour put Leicester back in front as Brendan Rodgers' team looked to take home a crucial three points in the Champions League race.
Man United were unable to find another goal, though, as Leicester moved up to third place on 66 points from 36 games, two ahead of fourth-placed Chelsea who have a game in hand.
"It was a huge step [towards the Champions League]. I am so proud of the team," said Rodgers, whose side face Chelsea in the FA Cup final on Saturday.
"The players have definitely overachieved with all the big teams around. Now we can really enjoy the final at the weekend."
After the match, United coach Solskjaer said his team would need to improve to be on the level of champions City.
Just moments after rivals Manchester City secured the Premier League title, Manchester United manager Ole Gunnar Solskjaer sent a clear message to the club's hierarchy.
"If you want to take up the challenge with them we need to strengthen our squad," Solskjaer said. "The whole season has given us optimism and room for optimism but also improvement.
"Congratulations to [City]. They have had a fantastic season. I am very pleased with my boys that have made it to the last 10 days with a Manchester City side ahead of us that are lauded as the best in Europe.
"We want to take the next step and it might be that we need a couple more to strengthen the squad."
Information from Reuters was used in this story.