Second-half goals by Dindane and Hermann Hreidarsson settled a nervy encounter at Fratton Park as Pompey claimed only their third Premier League win of the season.
Ivory Coast striker Dindane had earlier seen his poor penalty saved by Burnley goalkeeper Brian Jensen after wrestling the ball from regular spot-kick taker Kevin-Prince Boateng.
Dindane was only given permission to play on the morning of the game after parent club Lens agreed to wave a clause in his loan deal that cash-strapped Pompey had to buy him permanently once he had made 12 appearances.
Pompey's top-scorer must have been wishing he was sitting in the stands after spurning a golden chance to open the scoring after only 11 minutes.
Dindane wrongly assumed he was offside and allowed the ball to trickle harmlessly under his foot and wide of Jensen's left-hand post.
Burnley striker Steven Fletcher produced an equally glaring miss six minutes later, completely missing his kick after being played through by Wade Elliott.
Portsmouth, who could not afford to pay their players on time last month, were presented with a glorious chance to take the lead 13 minutes before the interval when Elliott was harshly penalised for a foul on Hreidarsson.
Referee Phil Dowd pointed for a penalty but TV replays showed Hreidarsson dived and Burnley felt justice was done when Dindane sent a weak spot-kick straight into the arms of a grateful Jensen.
Angry Portsmouth fans booed their team off at half-time and only two superb saves by goalkeeper Asmir Begovic, who acrobatically denied Kevin McDonald and Fletcher, stopped the Clarets going in front.
The hosts took the lead against the run of play in the 65th minute.
Boateng and substitute Kanu combined neatly on the edge of the penalty area before the veteran Nigerian picked out the onrushing Hreidarsson with a clever flick.
Hreidarsson kept his composure before arrowing a right-foot shot across Jensen and into the bottom left corner.
Dindane sealed the win six minutes from time with a well-taken back post header from Jamie O'Hara's cross.