Paul Konchesky denied the hosts a much-needed cleansheet with a dipping drive after 20 minutes, with Ross Wallace's header restoring parity three minutes later.
The tireless Nugent scored what proved to be the winner shortly after the break and Paul Gallagher's penalty added gloss to a spirited performance.
The victory should go some way to persuading the kind of high-calibre manager desired at the King Power Stadium that the Midlands club can still challenge for promotion this season.
Nugent lit the fuse of a feisty encounter, testing Lee Grant with a speculative drive from the edge of the area that the home goalkeeper did well to parry.
Kasper Schmeichel's goal-kick was then nudged onward with a well-judged header from Steve Howard, Grant shaping to repel the ex-Preston striker's attempt but seeing the ball sail over the crossbar.
A moment of uncertainty from Sol Bamba allowed Burnley to take the initiative, Jay Rodriguez stealing possession from the French defender but allowing Schmeichel to gather easily.
Yet ultimately it was Leicester who were rewarded for their early intent, albeit through an unlikely source. With 20 minutes played Konchesky, a summer signing from Liverpool, appeared on the left flank to seize a loose ball. He glanced up before scooping his shot against Grant's fingertips and into the top corner.
Burnley's response was immediate and when Bamba's careless clearance bounced high and onto the head of Wallace, he instinctively nodded the ball over Schmeichel and into the gaping net.
Eddie Howe's charges threatened a second from the restart, Marvin Bartley slipping past an ailing Bamba but clipping tamely against Schmeichel.
And in the 37th minute, Charlie Austin pounced on a sublime through-ball from Rodriguez but dragged his strike against the upright and wide.
The Foxes were perhaps unlucky to go in at the break with 11 men as Neil Danns avoided sanction for a late challenge on Bartley.
Andre Amougou saw his header cleared off the goal-line by Lee Peltier as play resumed.
But it was former England cap Nugent, impressive throughout the first half, who had the visitors ahead in the 54th minute.
Gallagher stormed down the right wing before supplying a pinpoint cross for Nugent to head home gracefully from close range.
The provider turned scorer eight minutes later. Referee Geoff Eltringham adjudged Amougou to have body-checked Andy King in the penalty area, with Gallagher on hand to confidently power home the resulting spot-kick.
As the atmosphere soured at Turf Moor, so tempers began to flare on the pitch.
Amougou went in late on Nugent and Eltringham resisted strong calls to sanction Bamba after a clumsy clash with Kieran Trippier floored the Manchester City loanee.
Substitute Keith Treacy found Schmeichel's gloves as the game entered the final 15 minutes.
Austin, eager to fire a third goal in as many league matches for the Clarets, twice produced wayward headers inside the six-yard box as the hosts sought to find a way back into the match.
With two minutes remaining, Rodriguez might have provided further consolation but his snapshot was blocked by Peltier.