Nugent struck shortly after the half hour as the Clarets' afternoon went from bad to worse.
The hosts had already received a hammer blow when leading scorer Charlie Austin, having passed a late fitness test on a muscle injury, was replaced by Martin Paterson after a mere eight minutes.
And it was testament to Burnley's battling spirits that the Clarets made a real game of it despite the loss of their star man.
In fact, Burnley made the brighter start of the two sides and Chris McCann tried his luck from distance after only 11 minutes, only to see his effort go wide.
At the other end, Leicester's Ben Marshall twice came close, first seeing an effort blocked by Kieran Trippier and then bringing the best out of Burnley keeper Lee Grant who tipped his 17th-minute effort round the post.
Paterson brought a save out of Kasper Schmeichel as the Clarets went toe-to-toe with their promotion-hunting rivals, while Leicester's Lloyd Dyer also tried his luck in an end-to-end start at Turf Moor.
The Foxes went into the game without a goal in three matches, but boss Nigel Pearson's long wait was finally over after 31 minutes when Nugent struck home from close range following an assist from Wes Morgan.
The lead could have been doubled just three minutes later but McCann came to the rescue, blocking an 18-yard effort from Anthony Knockaert.
Burnley had taken time to recover their cool after conceding, but eight minutes from the break Danny Ings just cleared the bar from inside the box to raise home fans' spirits.
Even so, the Clarets could have conceded a second minutes from half time when Dyer again worked himself a good shooting opportunity, only to lift his 12-yard effort over the crossbar.
The Foxes picked up where they had left off after the interval as Matty James saw Jason Shackell get in the way of his effort as Pearson's side turned the screw at the start of the second half.
But Schmeichel had to be at his alert best to thwart Paterson an equaliser from point-blank range after 59 minutes as Sean Dyche's men sensed at least a point was still there for the taking.
The Leicester keeper was again in the thick of the action with 20 minutes left, this time comfortably saving an on-target Ings effort from 18 yards.
Livewire Ings then saw an effort blocked by Liam Moore as Leicester were forced back on the defensive, Schmeichel then tipping another Paterson attempt round the post as the Clarets battled to the end.