PSG, who had been expected to challenge for the Champions League places but languish down in 11th in Ligue 1, secured a berth in next season's Europa League with this eighth Cup triumph.
Arch-rivals Olympique Marseille, who are about to win their first Ligue 1 title since 1992, are the only club with more French Cup triumphs than the capital side as they have won it 10 times.
"We were under a tremendous pressure, with lots at stake as we had no choice but to win to salvage this very bad season," PSG coach Antoine Kombouare told reporters.
"It kind of makes up for all the rubbish we've done this season," he added.
PSG dominated throughout the game but Monaco keeper Stephane Ruffier denied them until extra time.
Ruffier was left helpless when Hoarau headed home the winner into an empty net from close range just before the extra-time interval.
"We fully deserve this win because my players played a great game... I wish they had played like this during the whole season," Kombouare said.
"I'm very proud of them because you cannot imagine how hard it is for them to play for PSG with this huge pressure from the fans and the media.
PSG fans heavily outnumbered their Monaco counterparts at the Stade de France and it was the Parisians who created the first clear chances.
Defender Mamadou Sakho first had a volley from a Stephane Sessegnon corner cleared off the line by Juan Pablo Pino. Ruffier then made his first brilliant save of the night to deflect a superb Ludovic Giuly shot that was going into the top corner in the 15th minute.
The Monaco keeper kept out a close range effort from Mevlut Erding before the half hour after the Turkey striker had been set up in the box by a defence splitting pass from Christophe Jallet.
Erding wasted another chance when he then sent a low shot wide from inside the box.
Monaco waited until just before half time to threaten PSG goalkeeper Apoula Edel who denied Pino the opener after the Colombian striker had dribbled into the box.
"Playing a final before 70.000 PSG fans is a bit difficult for a young team such as mine... When you play in such a hostile atmosphere, it makes a little difference for Paris," Monaco coach Guy Lacombe told reporters.
PSG kept much of the possession after the interval and had penalty claims waved away by referee Lionel Jaffredo when Erding tumbled over.
Erding squandered a last opportunity to avoid extra time when his shot bounced off the bar although he was alone in front of Ruffier in the 82nd minute.
PSG were finally rewarded for their domination in extra time when Hoarau nodded a rebound into the empty net after Ruffier had made another save from a powerful Jallet shot.
