The Foxes took a deserved lead through Konchesky when he launched a thunderous strike beyond goalkeeper Lee Grant.
However, the home side levelled soon after when Ross Wallace headed in after a mistake from Kasper Schmeichel.
Nugent then headed home a Gallagher cross, before Gallagher got his name on the scoresheet from the penalty spot.
A superb diving save from Grant kept out Nugent's long-range strike early on, while at the other end Jay Rodriguez found himself in acres of space but could only shoot straight at Schmeichel.
Did you know?
Before their defeat by Leicester, Burnley were unbeaten in 28 home midweek night games since QPR won at Turf Moor in December 2007
Konchesky grabbed his first goal for the Foxes after 20 minutes when he latched onto a poor defensive header to rifle the ball past the Burnley keeper.
The lead was short-lived when just three minutes later a poor punch by Schmeichel fell straight to Wallace, who headed into an empty net.
Charlie Austin could have given the home side the lead before the break when he latched onto the end of a Rodriguez through-ball, only for Sol Bamba to get the slightest of touches to direct the shot onto the bottom of the post.
The lively Nugent, who scored six goals for the Clarets in the Premier League, had two chances go begging before his persistence paid off in the 54th minute, when former Blackburn forward Gallagher launched an inch-perfect cross to the back post for Nugent to head home his fifth goal of the campaign.
Gallagher extended Leicester's advantage from the penalty spot just after the hour mark, after Richie Wellens had jinked his way through the Burnley backline only to be obstructed by Andre Amougou.
Burnley pressed late on with Austin having two headers in quick succession go narrowly wide of the post, before Rodriguez's late effort was cleared by Lee Peltier.
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