The League One side, who have never progressed beyond the third round of the competition, held their own against a much-changed Foxes outfit in Lancashire.
Leicester, who were unable to call on the injured Jamie Vardy against his former club due to a groin injury, did not manage a shot on target during the game in his absence.
Indeed, the Cod Army came close to snatching a dramatic victory in stoppage time when Ashley Hunter's deflected strike came off the post.
The replay will be played in the week commencing January 15 at the King Power Stadium, while the fourth-round draw will take place on Monday, January 8.
FLEETWOOD TOWN (4-3-3): Neal; Coyle, Bell, Eastham (Pond 14'), Bolger; Dempsey, Glendon, Schwabl; Burns (McAleny 68'), Hunter, Cole
LEICESTER CITY (4-3-3): Jakupovic; Amartey, Dragovic, Benalouane, Chilwell; King, Silva (Okazaki 57'), James; Barnes (Albrighton 57'), Gray, Slimani (Iheanacho 79')