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Leicester might have found themselves ahead after just three minutes, but Dean Hammond was unlucky as he saw his header fly over having climbed highest from a corner.
As Bury continued to weather an early storm, Leicester forward Andrej Kramaric sought to take matters into his own hands, only for Walton to produce another standout stop to deny his counterpart in the 20th minute.
Debutant frontman Dodoo soon came to the rescue, though, as the rising striker collected in the middle of the box to produce a memorable finish into the top-left corner.
The away side put themselves in an even better position just before half time, as Kramaric returned to register having been picked out by strike partner Dodoo.
Leonardo Ulloa had a quiet first half for the Foxes and his frustrating evening worsened when he spurned a chance to put the game to bed, the striker guiding his shot high and wide from the centre of the area.
Thankfully for the visitors, Dodoo capped off an impressive evening as he sealed a brace four minutes from time, before going on to complete an impressive hat-trick to secure Leicester's passage into the third round.