The Foxes’ hero was in-form striker Kelechi Iheanacho, who struck twice and now has nine goals in nine games.
He took advantage of Fred’s awful backpass to round Dean Henderson to score in the 24th minute after the helpless Brazilian was played into trouble by captain Harry Maguire.
Mason Greenwood equalise for united before half-time but Youri Tielemans restored Leicester City’s lead with a fine low shot seven minutes after the break before Iheanacho took advantage of shocking marking at a free-kick to score Leicester’s third goal in the 78 minutes.
United introduced Bruno Fernandes, Edison Cavani and Luke Shaw and maintained some pressure, but it was not enough as Iheanacho’s back-post header sealed the game with barely minutes to go.
The semi-final against Southampton will be held at Wembley on the weekend of April 17-18.