Jonny Evans turned the ball into his own net under pressure from Clarets goalscorer Chris Wood with 10 minutes left, but the VAR review suggested the striker had tripped the Leicester defender as they both tried to get to the ball following a save from Kasper Schmeichel.
With the contact seemingly minimal, Burnley boss Sean Dyche cut a frustrated figure on the sidelines as his side were unable to fashion a leveller thereafter.
The visitors were in front via Wood midway through the first half before Jamie Vardy levelled on the stroke of half-time.
Leicester were not at their fluent best on the day but Tielemans crashed a shot in off the underside of the crossbar on 75 minutes to give Brendan Rodgers’ side all three points.
The win moves Leicester into second spot on goal difference, at least until after Manchester City’s trip to Crystal Palace later this evening.
Both teams had chances early on but it was Wood who made the first incision.
He picked up possession from a long ball and released Dwight McNeil before getting between Evans and Caglar Soyuncu inside the box to power a header into the net, despite Schmeichel getting a hand on the ball.
Tielemans almost levelled soon after but his shot was parried away by Nick Pope, while Vardy uncharacteristically hit the side-netting having got away from the Burnley cover minutes later.
Vardy had his goal in the final minute of the half, heading home a cross from Harvey Barnes.
Wood threatened again shortly after the break, while Tielemans and James Maddison had efforts on the Burnley goal as Leicester upped the ante after their fans paid an emotional tribute to former owner Vichai Srivaddhanaprabha on the hour.
The home side hit the front entering the final 15 minutes, with Tielemans coming up with a fine finish following good work from Andre Gray.
The drama was still to come, however, with Evans’ moment of misfortune coming after Schmeichel had made a decent stop.
Burnley boss Dyche is unlikely to let the VAR decision pass silently but Rodgers and Leicester will gleefully accept the three points.