The capital side, who owed their victory to Issa Cissokho's own goal a minute into added time in an otherwise dull final, claimed the title for the third year running and have now won 32 consecutive League Cup and French Cup games.
Their last defeat in a cup match came in the 2013-14 season when they were knocked out by Montpellier in the last 32 of the French Cup.
The result should ease the pressure on coach Unai Emery after his side flopped in the Champions League and lost their Ligue 1 title to Monaco.
"It was not an easy season but PSG have what it takes to get a major trophy soon," said PSG sporting director Patrick Kluivert.
"It's good to win yet another cup. Credit to Angers, they did what it takes to resist," said PSG midfielder Blaise Matuidi.
PSG, who now own one more French Cup title than Marseille, got off to a solid start but Alexandre Letellier denied Edinson Cavani and Marco Verrati early on.
Emery's side struggled to threaten and Angers came closest to scoring midway through the first half when Nicolas Pepe's fierce 20-metre strike hit Alphonse Areola's post.
Alexandre Letellier made a fine save in the 56th minute to deny Matuidi as Angers held firm.
Six minutes from time Cavani, who has scored 49 goals in all competitions this season, squandered a chance to match Zlatan Ibrahimovic's 2015-2016 tally of 50 when he missed the target from close range.
But PSG eventually sealed their win when Senegalese Cissokho put through his own goal when trying to defend a corner.
"I have another year's contract and I am happy here. I think this team can grow," said Emery.
The result earned Girondins de Bordeaux, who finished sixth in Ligue 1, a Europa League spot next season.
(Reporting by Julien Pretot, editing by Pritha Sarkar)
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