The ex-England striker came off the bench to secure the points for the Foxes after fellow substitute James Maddison had broken the deadlock at Turf Moor.
The 35-year-old netted the seventh goal of his career against the Clarets after Harvey Barnes had put the ball on a plate for the forward in the 90th minute.
The hosts were just minutes away from climbing out of the bottom three, at the expense of Everton, but a double-change from Brendan Rodgers altered things significantly.
Vardy set up Maddison in the 82nd minute and then rounded things off when heading home his first Premier League goal since November.
Burnley goalkeeper Nick Pope had previously done everything he could to keep the visitors out — keeping out Ricardo Pereira, Patson Daka and Harvey Barnes twice — during a display that ought to enter him into the discussion for England selection.
Here are the ratings.
5. Ben Mee 6
Determined and dogged, the Burnley skipper defended his penalty area with authority. Got his head on anything airborne, and put his foot through the ball to clear when low-driven crosses posed a threat. Sustained an injury in a typically robust challenge with Daka and was replaced by Collins just before the interval. Photo: Alex Livesey
6. Aaron Lennon 7.5
The 34-year-old has had a new lease of life of late. Earned applause for several lung-busting runs to arrest Leicester counter attacks, though the one to halt the troublesome Barnes after the break was the pick. Didn't stop running at Thomas, giving the young full back a real test, and unfortunate not to claim an assist when Cornet had strayed marginally offside. Photo: Mike Hewitt
7. Dwight McNeil 6
Made an impression in fits and starts. Quiet in the first half when unable to get beyond Pereira, who got a shot off when turning McNeil far too easily in the ninth minute. Came to life after the break, with a little sprinkling of magic carrying the Clarets into threatening areas of the pitch, but lacked some finesse with the end product. His pass for Cornet, however, deserved a finish. Photo: Alex Livesey
8. Josh Brownhill 5.5
Always willing to join his team-mates in the trenches, knuckling down and doing the dirty work in his attempts to negate the threat of Leicester's front three while trying to dilute the influence of Tielemans. Produced a vibrant display, never stopped running, but never really a match for Ndidi physically and guilty of several stray passes. His crossfield pass for Roberts in the second half, however, deserves a special mention. Photo: Nathan Stirk