The young Foxes hosted the Citizens at LCFC Training Ground on Saturday afternoon and went ahead via Tommy Neale's 28th-minute goal.
The visitors' equaliser arrived shortly after the hour-mark, courtesy of Jaden Heskey, taking the game to an additional 30 minutes in Seagrave.
Just before the interval in extra-time, Ashton Muir found a way through for Man City and Matthew Warhurst made sure it would be the Manchester club progressing into the final late on.
Leicester return to U18 Premier League South action next Saturday, facing West Bromwich Albion at LCFC Training Ground (11am GMT kick-off).
The details
City: French; Aluko, McAlinney (Khela 70'), Amartey, Ali (Donnelly 63'); Neale, Joseph; J. Evans, Ononaye, Otchere (Thomas 101'); R. Evans (Toure 79').
Sub not used: Bausor.
Goal: Neale 28'.