Patrick Bamford converted a fifth minute penalty while the Clarets had a 19th minute equalizer by Ashley Barnes ruled out in a game where both teams had good chances.
Story of the match
The decisive moment of the match came in the fourth minute when Nick Pope brought Bamford down and a minute later the striker fired into the top corner, sending Burnley goalkeeper Ben Mee the wrong way.
The Clarets thought they had equalized in the 19th minute, Meslier jumping into Mee, dropping the ball and Ashley Barnes fired into an empty net only to see a foul called on the Whites' goalkeeper.
Burnley's defense was continuing to come under pressure, Nick Taylor blocking Bamford's shot and Mateusz Klich going over from 25 yards out. Pope then blocked Jack Harrison's low cross.
The Burnley goalkeeper then nearly gifted the hosts a second, giving away possession to Raphinha, who cuts it back to Harrison and his shot hits Bamford and bounces away, a let-off for the Clarets.
In the 34th minute, Luke Ayling came to Leeds' rescue after Chris Wood sent a header in from Ashley Westwood's cross that the defender cleared away with his head. Three minutes later, Erik Peters bounced a volley wide of the upright.
Raphinha then had a go as took advantage of Mee's poor defensive header, dropping it into the far corner, but it was too close to Pope, but the Burnley netminder saved comfortably.
A minute before halftime, Harrison and Rodrigo played a neat one-two that the latter sent towards Pope, who had no trouble saving his header.
Leeds' defense was put under pressure early in the second half, Kalvin Phillips sparing the Whites' blushes first getting his head to the ball before Wood then blocking Mee's flick.
Meslier then made a double save from Matthew Lowton and Josh Brownhill while Pope dove to his left, tipping Pablo Hernandez' shot around the post.
Ten minutes from time, James Tarkowski made a crucial intervention to deny Bamford after Hernandez had cut the ball back from the byline.
Jay Rodriguez had the visitors' best chance of the game in the 82nd minute as he was on his weaker left foot and on a tight angle, but skied it from close range.
Meslier made one final save after Tarkowski had fed Ashley Barnes, who blasted a shot from an angle that the Leeds keeper saved at his near post.
Man of the match: Ilian Meslier
The Leeds defense was shaky at best throughout the game and it was up to the Frenchman to rise to the occasion and he did. While he was involved in the controversy that denied Burnley an equalizer, he continues to show why the club made him their number one goalkeeper.
Leeds' defense needs to improve
As just mentioned, the Whites' back line was far too generous in giving up quality chances to the Clarets and if not for the heroics of Meslier, they would have been exposed. While the set piece issue that usually troubles them didn't, they need to sort their defense as quickly as possible.