In a game between two sides in search of a manager, it was the Clarets' prolific striker Andy Gray who did the damage in the 23rd minute when he was fed by Wade Elliott and turned Patrick Kisnorbo before beating Marton Fulop with a crisp angled drive for his tenth strike of the season.
The result left City without a win or goal in the five games they have played against Burnley at the Walkers Stadium since vacating their old home Filbert Street.
Clarets keeper Gabor Kiraly was not called into action until the 30th minute when he comfortably saved Matt Fryatt's tame shot and it did not improve after the break for the toothless Foxes, despite having four strikers on the pitch for the final half-hour, as they failed to muster a chance worthy of the name.
And, but for wayward finishing, the scoreline would have been more embarrassing for the home side.
Midway through the second half, Fulop saved City's blushes when he turned aside Burnley substitute Alan Mahon's shot for a corner which Leicester were relieved to clear.
With 15 minutes to go, Robbie Blake found Kyle Lafferty free on the edge of the box and, but for a smart save by Fulop, Burnley would have doubled their advantage.
Despite fielding four fowards for the final 30 minutes, City's best retort was a weak shot from Dudley Campbell and a wayward drive from Stephen Clemence.
Such was City's raggedness that the loudest applause of the day was reserved for the half-time parade by ex-servicemen and the British Legion with the home side having to troop off at the end to a chorus of boos.