Ledesma, the match winner at Millwall last time out, got the first-half winner as Boro clocked up back-to-back wins for the first time in 2013.
Boro's first effort came after four minutes as Ledesma curled the ball into the box but Albert Adomah sliced his shot wide.
Then after six minutes Adomah wriggled down the left, cut inside and chipped to the far post where Lukas Jutkiewicz sent a glancing header just wide.
Boro continued to press and after 10 minutes Ledesma sent a low effort from distance wide of the far post then soon after put a free-kick dipping just over the bar.
Then Ledesma floated a free-kick into the box for Jonathan Woodgate to head straight at the keeper.
But Boro's pressure told after 24 minutes as the livewire Ledesma cut in from the right to drill in a low 25-yard angled effort that skidded towards the near post and bounced over diving Burnley keeper Tom Heaton.
Burnley clawed their way back into the game and almost levelled after 40 minutes as Dean Marney crossed from the right for Danny Ings to power in a header from eight yards that Shay Given tipped over superbly.
Boro went close after 55 minutes as Marvin Emnes touched inside for Jutkiewicz to drill in a shot that the keeper did well to turn over.
Boro continued to press and went close on the hour as Ledesma deftly chipped the ball over his shoulder then spun and darted away to lose two defenders.
He cut inside before lashing a low 25-yard effort wide of the far post.
Burnley had a good spell and both Woodgate and Daniel Ayala had to head away dangerous crosses then George Friend bravely charged down a piledriver from Junior Stanislas.
Boro threatened after 78 minutes as Curtis Main did well to trick his way down the line then slotted a pass between two defenders for Grant Leadbitter to feed the ball into the.
Jozsef Varga went flying under a challenge but with the crowd screaming for a penalty referee Mark Clattenburg booked the defender for diving.
Burnley went close after 80 minutes as Keith Treacy crossed and Marney somehow headed over from close range.
Burnley piled on the pressure in the closing stages and went close after 88 minutes as Stanislas turned just outside the box then drilled in a 20-yard effort that Given did well to tip over.