Marney received his marching orders for a tackle on Blues defender Paul Robinson with the game poised at 1-1 but the former Tottenham and Hull midfielder will now avoid a three-match ban and is available for this weekend's trip to Peterborough.
Clarets boss Sean Dyche told his club's official website: "It does not stop us being aggrieved at the decision, which clearly affected the outcome of the game when we looked good value to go on and get the three points, and I don't feel we have had a fair share of decisions go our way.
"But on this occasion the appeal decision has been a positive one for us and rightly so.
"I made it clear how I felt at the time, and after watching the incident again, and that view has been totally vindicated.
"We put our faith in pursuing the matter through the right channels and we are pleased that the process has gone in our favour.
"We are thankful that we have got a good player available to us to go forward and it's on to the next game to try and compete for another three points at Peterborough."