The Ligue 1 runners-up were frustrated for long spells in the capital until Issa Cissokho turned the ball into his own net in stoppage time to give Unai Emery's side their second trophy of the season.
There could hardly have been a greater difference in pedigree between the two clubs heading into the match, with Angers appearing in their first Coupe de France final for 60 years and PSG looking to continue their recent dominance of domestic cup competitions.
The capital outfit have now won each of the last six national cup competitions available to them, including the Coupe de la Ligue earlier this term, but their plans to make it a domestic double did not prove as straightforward as many might have expected.
Indeed, it was Angers who came closest to breaking the deadlock in the first half when Nicolas Pepe latched on to a rebound and fired a powerful effort towards goal, only to see his strike come back off the post.
Stephane Moulin's side were only promoted to Ligue 1 in 2015 and finished this season in 12th place, but they held on until half time to go into the break all square against their more illustrious opponents.
PSG came close to the opening goal shortly before the hour mark when Angel di Maria lined up a free kick and fired his effort narrowly wide of the target.
Angers continued to hold on, though, and the match looked to be heading for extra time when Edinson Cavani tried to catch the keeper out with a snapshot which flew past the post.
There was heartbreak for the underdogs at the death, though, as Cissokho inadvertently turned the ball into his own net when trying to clear a corner at the front post, only succeeding in glancing his header past the keeper.
There was no way back for Angers after that as PSG saw out the closing stages and sealed yet another trophy to give Emery a positive end to a largely disappointing debut season in charge.