David Nugent's contentious goal sent a weakened Portsmouth side into the fourth round of the Carling Cup.
Burnley had the best of the first half, Clarke Carlisle going close and Sylvain Distin blocking two Ade Akinbiyi shots.
Nugent, who missed a golden chance in the first-half when he tried to nutmeg keeper Gabor Kiraly, made no mistake on 69 minutes from Pedro Mendes' cross.
Burnley, who claimed Nugent controlled the ball with his hand, pushed hard for an equaliser but to no avail.
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