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.
13. Jay Rodriguez 5
Unfortunate to lose his spot in the side after building a promising partnership with Weghorst in recent weeks. Didn't have a great deal of time to put his stamp on the game when introduced for the final 20 minutes, but worked hard across the front and pressed well. Question marks over whether he could have done more to get a boot on Schmeichel's game-changing clearance. Photo: Alex Livesey