Angers defender Issa Cissokho suffered an own goal in PSG's corner kick into the stoppage time, reportrs Xinhua news agency.
PSG were level on 10 triumphs with Marseille before Saturday's final, where French President Emmanuel Macron featured and had a warm pre-game exchange with players.
Both sides and crowds in Stade de France observed one-minute silence for the victims in Manchester bomb attack Monday night.
Angers did pretty well in restraining the defending champions' attack, and their goalkeeper Alexandre Letellier finished several brilliant saves on PSG top scorer Edinson Cavani's attempts.
Angers also got their own chance of breaking the deadlock, as Nicolas Pepe's shot hit the post in a free-kick on 27th minutes.
The two teams were involved in a physical match-up, and PSG winger Angel Di Maria received a yellow card for altercation with his opponent.
After a goalless draw into the break, PSG still found it hard to beat tough Angers. In the 56th minute, Blaise Matuidi's close-range shot was blocked by Letellier in a quick response, and Di Maria's free-kick just went a little wide off the post two minutes later.
Cavani could not add another goal to his productive season (49 goals) as his shot went off the post against a defender in a counter-attack with six minutes remaining.
However, Matuidi forced Cissokho to head into his own net, finishing a last-gasp winner for PSG's historical victory in the tournament.
PSG won French League Cup and French Cup this season, while failing to achieve a domestic treble for the third straight year after losing the Ligue 1 title to Monaco.
They were also dumped out in UEFA Champions League last 16 by Barcelona in a humiliating manner after building a four-goal advantage in the first leg while losing the return leg 6-1 at Camp Nou.
(This story has not been edited by Social News XYZ staff and is auto-generated from a syndicated feed.)
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