The capital side missed a number of chances to take the lead in the opening 45 minutes, Edinson Cavani in particular guilty of wasting some fine opportunities, but the holders, who have now won the competition three years in a row, eventually got their goal in the 91st minute when Cissokho headed into his own net.
The victory somewhat saves the Parisians' disappointing season as they were dethroned as Ligue 1 champions by Monaco, while they were eliminated from the Champions League by Barcelona in the round of 16.
Unai Emery's men did win the Trophee des Champions and Coupe de la Ligue, though, and have now made it three trophies in the Spaniard's first season at the club.
PSG were high on confidence heading into the game, having beaten their opponents 2-0 on two occasions in Ligue 1 action this term, but things were not as straightforward this time around.
Stephane Moulin's side more than held their own and should consider themselves unfortunate following Cissokho's unlucky own goal in the dying seconds of the game.
PSG were in charge right from the off and Blaise Matuidi got an opportunity to open the scoring after just 12 minutes following a clever ball from Angel Di Maria, forcing goalkeeper Alexandre Letellier into a low save.
Cavani was next to threaten for Emery's men, but the Uruguay international aimed his header straight at Letellier, before being denied by the Angers goalkeeper a second time just minutes later after getting his foot to another intelligent ball from Di Maria.
Romain Thomas was then fortunate not to score an own goal when he slid Di Maria's low cross from the left against the upright, before clearing the ball for a corner.
Angers slowly grew into the game from there on and they came close to grabbing a shock lead in the 27th minute, Nicolas Pepe hitting the post with a superb half-volley from outside the area after the PSG defence failed to adequately clear the ball after a free kick.
Di Maria continued to impress after the break and the former Real Madrid star came within inches of breaking the deadlock on the hour mark when he curled a free-kick over the wall and into the side-netting.
PSG continued to dominate proceedings and Cavani should perhaps have scored the opener with five minutes left, shrugging off Ismael Traore after being sent through by Di Maria, only to then fire a low shot just wide of the target.
But there would be late drama after all when Cissokho inadvertently handed PSG the win in the 91st minute, beating his own goalkeeper with a clumsy header after Di Maria's corner.