The Nigerian broke the deadlock by pouncing on a horrendous mistake from Fred before Mason Greenwood restored parity for Manchester United with a sublime goal.
Tielemans put the hosts ahead once again early on after the break before Iheanacho popped up with a header in the 78th minute from a free-kick to seal the win.
Manchester United were listless all night and conceded goals in comical fashion. Even throwing on Edinson Cavani and Bruno Fernandes couldn't spark life into their insipid performance.
Leicester City deservedly went through and will now play their first FA Cup semi-final since 1982. On that note, here are the player ratings for Manchester United:
Dean Henderson - 5/10
Dean Henderson was beaten thrice on the night. Each time, it was someone ahead of him who made a comical error. Henderson wasn't at fault for any of the goals Manchester United conceded, though
Aaron Wan-Bissaka - 6/10
Perhaps the best defender for Manchester United on the night, but the bar couldn't have been any lower. Wan-Bissaka tracked Leicester City's attackers well and tried to nip their offensive plays in the bud. However, that came at a cost, as he didn't offer much at the other end.
Victor Lindelof - 6/10
While he deserves credit for his accurate passing, Victor Lindelof got everything else wrong. The Swede was thoroughly outclassed in defence and didn't complete a single tackle or interception.
Harry Maguire - 5/10
After a series of top performances, Harry Maguire came crashing back down to earth with a stinker. The Manchester United captain was all over the place at the back and lacked his usual silk and steel.
Alex Telles - 5/10
Alex Telles laid two key passes in the match, but that was about it for him. The Brazilian misplaced too many passes and left too many gaps at the back for Manchester United.
Fred - 4/10
It was an awful outing from Fred! He was sloppy in possession and was run over by Leicester City's marauding players. He also made a horrible backpass, which allowed Iheanacho to pounce on and open the scoring for Leicester City.
Nemanja Matic - 5/10
Nemanja Matic was unrecognisable because he was atrocious in every sense. The Serb was completely run over in midfield. Tielemans breezed past Matic as if he were a training cone before scoring Leicester City's second of the night.
Mason Greenwood - 6.5/10
Mason Greenwood seemed like the only Manchester United player who tried to get something from the game. He scored an unstoppable goal and then made a few more attempts to score as well.
Donny van de Beek - 5/10
While he did well with a dummy for Greenwood's goal, Donny van de Beek was largely anonymous on the night. The Dutchman couldn't bring any firepower to Manchester United's attack and didn't string a single key pass either.
Paul Pogba - 6/10
Paul Pogba assisted Greenwood and then attempted a nice shot at goal, which was saved by Kasper Schmeichel. But the Frenchman's impact waned considerably after the break as he became the first Manchester United player to be taken off.
Anthony Martial - 5/10
It was another flat performance from the Frenchman. Anthony Martial barely offered anything going forward.
Ratings of Manchester United Substitutes:
Edinson Cavani - 4/10
It was a desperate substitution by Ole Gunnar Solskjaer, but Edinson Cavani couldn't impact the game.
Bruno Fernandes - 5/10
Bruno Fernandes was another player the Norwegian manager turned to for goals. But he too returned empty-handed.
Scott McTominay - 4/10
Scott McTominay went into the books for bringing down Ayoze Perez and then failed to deal with Marc Albrighton's cross, which was turned in by Kelechi Iheanacho, who was lurking right behind him.
Amad Diallo - N/A
Amad Diallo came on too late in the game to merit a rating.