Matej Vydra gave the Clarets the ideal start when scoring his first league goal of the campaign.
The striker pounced on Hamza Choudhury's error, took the ball past Wilfred Ndidi before beating Kasper Schmeichel with an emphatic finish.
The Foxes equalised in the 34th minute when Kelechi Iheanacho hit Ndidi's pass over the top first time to beat Nick Pope.
Both sides had chances to win it with Schmeichel clawing James Tarkowski's header off the line before the break and then producing a wonderful fingertip save to keep out Chris Wood in the second half.
Ashley Westwood also saw his volley come back off the upright while Youri Tielemans was denied by the woodwork at the other end.
1. Nick Pope 6.9
Question marks over his positioning for the goal, but the Burnley stopper made crucial saves to keep out Choudhury and Tielemans in the first half.
2. Matt Lowton 7.4
Looked in a spot of bother when Castagne got down Burnley's right hand side at pace early on, but responded well. Tight at the back, carried the ball forward confidently and produced a number of pinpoint deliveries.
3. Charlie Taylor 7
Got forward well in support of McNeil down the left hand side and his deliveries caused issues in the Leicester box. Kept things tight down his side of the pitch and made a vital block to divert Pereira's strike off target in the second half.
4. James Tarkowski 6.8
Defended well when the Foxes changed shape and started to crank up the pressure. Made some good interventions when Leicester tried to play their way into the box and unfortunate not to beat Schmeichel with a first half header.