Heung-Min Son fired us into the lead with a great strike from the edge of the box six minutes before half-time.
Nacer Chadli completed the scoring in a second period that saw Michel Vorm make good saves from Demarai Gray and Marc Albrighton as we secured a deserved win.
Colchester United await in the fourth round a week on Saturday.
Leicester (4-3-2-1): Schmeichel, Simpson (Albrighton 63), Wasilewski, Benalouane, Chilwell, Dyer (Vardy 74), King, Inler, Drinkwater (Okazaki 46), Gray, Ulloa. Substitutes (not used): Schwarzer, Morgan, Kante, Mahrez.
Spurs (4-2-3-1): Vorm, Walker, Dier, Wimmer, Davies, Carroll, Bentaleb, Chadli (Onomah 73), Eriksen, Lamela (Kane 60), Son (Alli 84). Substitutes (not used): Lloris, Trippier, Vertonghen, Winks.
Goals: Spurs – Son 39, Chadli 66.
Yellow cards: Spurs – Davies. Leicester - Albrighton.
Referee: Anthony Taylor.