The Clarets passed a major test of nerve to keep on track for a possible European tour as they held off the challenge of the chasing Foxes to make it five Premier League wins in a row.
Early goals in the space of four minutes from top scorer Chris Wood and Kevin Long – his first in the Premier League – gave Sean Dyche’s men a flying start as they looked to open up a nine-point gap over Leicester in seventh place in the table.
That will be enough to secure European football for the first time in 50 years should Southampton not go on to win the FA Cup.
But a prize of that magnitude does not come easily.
And the Clarets had to show all their resolve and battling qualities to cling onto a priceless victory as Leicester, chasing a third successive away win, let off the shackles and gave everything to mount a fightback.
Their pressure paid off when the ever-dangerous Jamie Vardy pulled a goal back in the 72nd minute with his 20th of the season.
But Burnley refused to buckle further and with goalkeeper Nick Pope – again watched by England boss Gareth Southgate - continuing to grow in stature with another commanding display behind his tireless back four Dyche’s men had enough in hand to secure one of their most significant wins of an increasingly memorable campaign.
The Clarets could not have made a better start with two goals inside the opening 10 minutes.
Wood had scored at Turf Moor for Leicester four years ago with a goal that helped the Foxes clinch the Championship title ahead of Burnley.
But this time the damage was inflicted the other way as he notched his fifth goal in as many games and went into double figures for the season.
His partnership with Ashley Barnes had propelled the Clarets along their recent winning run and the pair clicked again when Barnes intercepted Wes Morgan’s attempted ball from the back and turned to slip in Wood with a neat reverse pass.
Wood’s first-time effort was saved by former team-mate Kasper Schmeichel but when the ball bounced back out, the striker showed a sure first touch to tee up a neat left-footed finish into the empty net from a diminishing angle.
Things then got even better when Long nodded in his first Burnley goal since January, 2014.
The return of Johann Berg Gudmundsson from a two-match injury absence had been the only change to the Clarets’ starting line-up and it was a deep corner from the Icelander that Long met at the far post with a header into the top corner.
Only a fine piece of defending from Danny Simpson prevented Wood making it three as he waited to connect with a cross from the marauding Matt Lowton.
But despite their early dominance, the Clarets would not have it all their own way for the remainder of the opening half.
With nothing to lose and the pressure possibly lifted from their shoulders, the Foxes then played with a freedom that brought them a handful of chances.
Vardy and Riyad Mahrez are players capable of turning any game and both had heading opportunities to get their side back in the game.
Vardy was guilty of missing the target at the back post but Mahrez connected well with a cross from Demarai Gray with a downward header which Pope did well to save low down to his left.
Mahrez provided one more moment of concern with the last kick of the first half, kissing the outside of a post with a swinging free-kick from 30 yards, and Leicester then made a change at the break by sending on Kelechi Iheanacho in place of Shinji Okazaki to try and increase their goal threat.
Burnley’s response was impressive as they regained a measure of control after the break and Gudmundsson was close to a third goal with a fierce drive which rose just over the top from 25 yards.
But Leicester responded again to push the Clarets back and Pope again needed to show his class with an excellent save at full stretch to turn a skidding effort from Vardy around the post.
It was last-ditch stuff from the Clarets at times as they fought tooth and nail to preserve the breathing space of a two-goal lead, only sporadically threatening at the other end – Barnes hitting a volley over the top from a James Tarkowski cross.
But with 18 minutes to go the Clarets’ defence finally cracked, Iheanacho threading a ball through for Vardy to crash the ball past Pope with a typically explosive finish.
Jeff Hendrick was sent on by Dyche to provide an extra body in midfield to attempt to stem the tide and the Irishman did his job as Leicester’s chances dried up.
The Clarets had to negotiate four minutes of added time but once more got over the line to seal a seventh home win and get within touching distance of another major milestone in the club's history.
Burnley: Pope, Lowton, Long, Tarkowski, Ward, Lennon, Cork, Westwood, Gudmundsson, Wood (Hendrick 75), Barnes (Vokes 85). Substitutes: Heaton, Taylor, Marney, Wells, Bardsley.
Booked: Lennon, Barnes
Leicester City: Schmeichel (Hamer 86), Simpson, Morgan, Maguire, Chilwell, Gray (Diabate 64), Silva, Choudhury, Mahrez, Okazaki (Iheanacho HT), Vardy. Substitutes: Albrighton, Dragovic, Fuchs, Barnes.
Referee: Martin Atkinson