Cavani headed home in the 65th minute to hand PSG their third trophy of the season, adding to their Ligue 1 and League Cup triumphs.
It was the capital club's ninth French Cup title and moves them within one of the record held by bitter rivals Olympique de Marseille.
PSG were without the suspended Javier Pastore and the Argentine playmaker was missed as Laurent Blanc's side looked short of ideas at a packed Stade de France.
Blanc decided to pick Brazil centre back Thiago Silva, who had been doubtful with a thigh injury, but his players were not at their best and Zlatan Ibrahimovic was a pale shadow of the brilliant forward he can be.
PSG's first clear chance came five minutes from the interval when Thiago Motta's fierce strike smashed the post against Auxerre who finished the Ligue 2 season in ninth place.
The French champions finally broke through 20 minutes into the second half when Cavani nodded Gregory van der Wiel's cross past Donovan Leon.