Sidibe gleefully finished from close range after strike partner Ricardo Fuller neatly rounded Michael Duff and crossed to the far post.
Neither side created much in front of goal although Gifton Noel-Williams twice went close for the Clarets.
Fuller had a 20-yard shot well saved by Mike Pollitt and Sidibe nearly grabbed a second with a fine curling effort.
Burnley boss Steve Cotterill:
"When you get a few moans and groans, they're frightened to accept the ball and we have to bully ourselves into accepting the ball sometimes.
"I thought Ricardo Fuller was excellent and he was the difference. If we'd had an on-song Ade Akinbiyi, maybe he might have been the difference. We had too many players that were off-colour."
Stoke manager Tony Pulis:
"Burnley is a very difficult place to come because they work hard and it's a fantastic result.
"I said to the players, after bad results, it's not taking the punch and getting knocked down, it's about getting up as quickly as you can and that's that we've done."
Burnley: Pollitt, Sinclair, Thomas, Duff, Harley, Elliott (Garreth O'Connor 71), Djemba-Djemba (Gudjonsson 66), James O'Connor, McCann (Lafferty 46), Akinbiyi, Noel-Williams.
Subs Not Used: Coyne, Foster.
Booked: Lafferty, Gudjonsson.
Stoke: Simonsen, Wilkinson, Duberry, Higginbotham, Griffin, Hoefkens, Russell, Matteo, Lawrence, Sidibe, Fuller.
Subs Not Used: Eustace, Sweeney, Pericard, Dickinson, Rooney.
Goals: Sidibe 24.
Att: 12,109.
Ref: M Pike (Cumbria).