The young Foxes were 1-0 up inside 20 minutes when Ethan Fitzhugh laid off Welsh forward Will Russ to score, but the hosts equalised through Oliver O'Neill, shortly before the interval.
Oliver Sanderson put Fulham ahead on 72 minutes, before Terell Pennant set up Kian Pennant to level the scores once more seven minutes later.
It was the Cottagers who took the lead again through Adrion Pajaziti's penalty and looked to be taking all three points from the clash at London School of Economics.
However, with just two minutes left to play, substitute Tawanda Maswanhise managed to find the net as City showed their powers of recovery to earn a point on the road.
Next up after the Christmas break for Leicester City Under-18s is an away encounter against West Ham United Under-18s on Saturday 9 January (12pm kick-off GMT).
City: Odunze; Godsmark-Ford (Wormleighton 64), Yfeko (Maswanhise 54), Obi; Marcal-Madivadua, Braybroke, Fitzhugh, Ewing (Cover 74); T. Pennant, Russ, K. Pennant.
Unused Subs: Doherty, Butterfill.
Goals: Russ 19', K. Pennant 79', Maswanhise 88'.