Caglar Soyuncu missed a good opportunity to give Leicester the lead before Chris Wood headed in Dwight McNeil's cross to put Burnley in front after 26 minutes.
Leicester responded well and equalised on the stroke of half-time after Vardy rose impressively to head in Harvey Barnes' whipped cross.
Vardy had a goal disallowed for offside but Leicester did take the lead on 74 minutes after Tielemans controlled Barnes' cross and finished into the roof of the net.
Leicester, who marked the first anniversary of the death of chairman Vichai Srivaddhanaprabha before kick-off, move up to third with 17 points.
Burnley slip to eighth place on 12 points.
Leicester: 25 Oct v Southampton (A)
Burnley: 26 Oct v Chelsea (H)