The Foxes, whose three away games have ended in defeat, were up against it when Dean Marney fired in an early opener to give Burnley the lead.
But David Nugent levelled matters with the second half only two minutes old, finishing from close range before Jamie Vardy scored his first home goal for the club with a neat finish to seal the points 10 minutes later.
The win sees Nigel Pearson's side climb nine places in the table and all of a sudden their outlook for the season looks more promising.
For Burnley, who hit the post late on, it was a third defeat on the road and with a trip to Derby coming on Saturday, boss Eddie Howe will be looking for a quick cure to their travel sickness.
Pearson made three changes to his Leicester side has he looked for a response from Sunday's defeat at Wolves, while Burnley were unable to call on Charlie Austin, who bagged a hat-trick at the weekend, as he was unable to shake off a knock.
The Clarets did not miss their top scorer too much as they burst into a 10th-minute lead.
It had been the hosts that had made the early running, but when Martin Paterson broke down the right, his cut-back was perfect for Marney to crash in his second of the season from 10 yards.
Leicester probed for an immediate response that their home fans demanded, but they came up against a Burnley defence in no mood to give anything away and Ben Marshall's effort from 20 yards, which stung the hands of Lee Grant, was the Foxes only moment of promise.
The hosts were eventually able to build a bit of pressure and they went close to levelling when Anthony Knockaert's deflected effort went just wide of Grant's post.
They came even closer shortly after when Nugent rattled the woodwork with a curling effort from distance as Burnley survived to the break with their lead intact.
But their survival did not last long into the second half as Leicester got their reward and it was Nugent who did the damage with his first goal of the season.
Marshall hurled a long throw into the area, Vardy flicked on and Nugent pounced to bundle home from close range.
And Leicester completed the turnaround 10 minutes later when Vardy got in on the action, turning from Knockaert's pass and firing in from a narrow angle.
Burnley searched for a leveller, but in doing so they left themselves open to counter attack and Grant had to be alert to keep out a Danny Drinkwater effort.
The visitors came within inches of levelling in the final 10 minutes when Sam Vokes latched onto a loose ball, but his snatched effort grazed the post.
But it was Leicester who finished the stronger and Knockaert and Lloyd Dyer were both off target as it ended 2-1.