David Nugent was Pompey's controversial hero as his goal proved the difference at Burnley.
Nugent's 69th minute effort gave the Blues a place in the Carling Cup fourth round.
But the home side were left furious as they claimed the striker handled the ball in the build up to the goal.
Referee Mark Clattenburg ruled he used his chest to control Pedro Mendes' through ball before clipping a cool finish past Gabor Kiraly.
Nugent's goal silenced the taunts of the Turf Moor crowd and those who were preparing to harangue the 6m signing after missing a gilt-edged first-half chance.
He was played clean through and broke Burnley's offside trap after 18 minutes but dallied as he tried to round Kiraly and allowed the keeper to gather.
Burnley fans hammered the striker thoughout because of his connections with local rivals Preston.
It was a game of few chances as Pompey rang the changes after the win at Blackburn.
Burnley huffed and puffed but Jamie Ashdown was never seriously extended.
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