Matt Taylor, making his first start since August, won a penalty for the home side and took it himself but he could only strike the outside of the post with Kapser Schmeichel beaten.
City promptly broke down the other end and when Marc Albrighton swung in a cross Michael Duff deflected the ball goalwards and despite Tom Heaton’s best efforts, Jamie Vardy was on hand to force home what proved the matchwinner (60).
The scoreline meant Leicester won four successive top flight matches for the first time since October 1966 and, after 140 days at the bottom of the table before last week’s win over Swansea, they are now clear of the drop zone by a single point, something not seen since November 17.
For Burnley the wait for a goal goes on as another blank means they are one minute short of eight hours since they last scored on 14 March.
That win over defending champions Manchester City now seems a long time ago as they have taken just one point from five matches and are now five points from safety.
Burnley had been dealt a blow before kick-off when striker Sam Vokes was ruled out by injury, handing Lukas Jutkiewicz his first Premier League start in six months.
The 26-year-old had more yellow cards (one) than goals in 25 previous Premier League appearances this season so it is safe to assume the Clarets were not pinning their hopes on any survival miracles from him.
Danny Ings saw an early shot blocked by Wes Morgan and the best other chance of the error-strewn first half dropped to Taylor but Schmeichel was equal to his close-range flick.
How pivotal the 90 seconds around the hour mark were may not yet be know but there was no doubt in the minds of the visiting supporters, who celebrated wildly after going ahead.
Taylor had been brought down by Paul Konchesky when Schmeichel parried a shot from Ings after the Burnley striker had jinked one way and then the other to create an opening.
Taylor, whose last goal was against the Clarets for West Ham in the Capital One Cup in October 2013, picked himself up to take the most pressurised of spot-kicks but – having sent Schmeichel the wrong way – he could only watch in despair as the ball hit the outside of the post and flew behind.
To add potential catastrophic injury to that insult, Leicester went straight down the other end where Marc Albrighton’s cross cannoned goalwards off Duff for Vardy to pounce on the goal line.
Burnley had one real chance to salvage a point but Schmeichel produced a fine reaction save after Robert Huth inadvertently turned Ben Mee’s cross goalwards.
Survival often hangs on fine margins and Leicester’s latest victory could yet yield far greater reward.
Phil Thompson’s Soccer Saturday verdict
Both teams did lack that quality today, both teams were launching it. It’s paracetamol all round because there were so many headers! It is just great drama, though. The three centre backs for Leicester – Wasilewski, Huth and Morgan – were absolutely fantastic. Four great victories for Leicester City; they weren’t as good as they were last week but they put in a shift.
Burnley: Heaton (6), Trippier (5), Duff (7), Shackell (6), Mee (6), Boyd (6), Jones (7), Arfield (6), Taylor (6), Ings (6), Jutkiewicz (5).
Subs: Wallace (5), Sordell (5), Kightly (5).
Leicester City: Schmeichel (7), Wasilewski (6) Huth (7) Morgan (8), Albrighton (7) King (6), Cambiasso (6), Drinkwater (6), Konchesky (6); Vardy (7), Ulloa (6).
Subs: De Laet (6), James (5), Kramaric (6).
Man of the match: Wes Morgan