Mason Greenwood levelled the score at 1-1 in the first half but Ole Gunnar Solskjaer’s side were unable to book a place at Wembley.
The hosts did take the lead less than 10 minutes later when Iheanacho pounced on a poor back pass from Fred to skip round Henderson and score into an empty net.
United equalised six minutes before half time when Greenwood got on the end of a Paul Pogba cross following a smart dummy from Donny Van de Beek.
Leicester restored their lead after 52 minutes when Tielemans drove at the heart of the United defence and slotted the ball into the bottom corner from the edge of the box.
Leicester almost grabbed a third minutes later but Vardy dragged his shot wide when one-on-one with the goalkeeper.