Burnley, who were one of the longest serving members of the Championship, opened the scoring at Wembley in the 13th minute with an exquisite strike from Wade Elliott.
Kevin Blackwell's side may have had a penalty but Owen Coyle's side had the better of the chances in the opening period.
In front of a crowd of 80,518, both sides had chances early in the second half with Montgomery denying substitute Gudjonsson in the 54th minute with a great block and another by Walker stopped Robbie Blake from scoring eleven minutes later.
Jamie Ward had only been on the pitch for eleven minutes after coming on as a substitute for Sheffield United. In the 79th minute, he got to a Matthew Kilgallon through ball but used his arm to control it and having already been booked was given a second yellow and then a red by referee Mike Dean.
The Blades were beaten and will face two matches against Coventry City in the Championship next season whilst Burnley go up.