Fulham struck first through Hugo Rodallega's close-range strike but two goals in the last four minutes of the first half from Wes Morgan and Chris Wood put Leicester in the ascendancy.
Ignasi Miquel's first Leicester goal gave them breathing space but Rodallega's header reduced the deficit within a minute.
Giorgos Karagounis' stunning free-kick looked to have forced extra time but Dyer's late intervention gave Leicester their eighth straight home win in all competitions and a place in the fifth round for the first time since 2000.
Leicester went in to the game with the best home record in the Championship and showed their intent after 35 minutes when Anthony Knockaert's fierce drive was spilled by Stekelenburg, who made amends by blocking the Frenchman's follow-up, while Paul Konchesky could only volley the rebound over from a tight angle.
And they got a deserved equaliser four minutes from the break as centre-back Morgan showed predatory instincts to nod home his first of the campaign after Stekelenburg spilled Knockaert's free-kick.
The game swung one way and the other within 46 second-half seconds. First, Leicester grabbed a third after 51 minutes when Drinkwater's near-post cross was converted by Miquel's bullet header but Fulham went straight up the other end and netted when Elsad Zverotic's deep cross was powered home by the head of Rodallega.
Fulham continued to press and were rewarded four minutes from time through a Karagounis free-kick that found Schmeichel's top corner.
