Following a goalless first half at Arbories Memorial Sports Ground, Lachante Paul put the Foxes ahead in the 47th minute with a smart finish across goal.
Annabel Blanchard, meanwhile, doubled City's advantage just three minutes later by curling it into the bottom corner from outside Burnley's penalty area.
Safe passage was confirmed on 61 minutes, meanwhile, with Paige Bailey-Gayle running onto a through ball to slot in Leicester's third.
Dani Cooper did get one back for the Clarets late on, a stunning strike from distance, but it is the Foxes who progress to the fifth round of the competition.
They will host Reading at Farley Way in the next stage of the Women's FA Cup on Sunday 16 February (2pm GMT kick-off).
The details...
City: Lambourne, N. Johnson, Morgan (c), Devlin (Everett 56'), Blanchard, Paul (Lawrence 60'), James, Thomas, Bailey-Gayle (Smith 70'), Rutherford, Plumtree
Subs not used: Field, Brown, L. Johnson, Popple
Goals: Paul 47', Blanchard 50', Bailey-Gayle 61'