The away side took an early lead after only five minutes through Alexandre Lacazette, who fired home with a standout finish past the Nantes keeper.
The hosts were back on level terms after 19 minutes, however, when Bessat broke free in the box to produce a smart shot into the bottom corner and make it 1-1.
Nantes then took the lead following a crazy two-minute period in which Bessat was on hand to grab another, as Michel Der Zakarian's side looked to pressurise the early cup favourites.
Lacazette came back to haunt the home side, though, and he created a telling ball to find Nabil Fekir in the 59th minute, the midfielder making no mistake with his effort to bring things level at 2-2.
Hubert Fournier's men might have thought that the worst of Bessat was behind them with the game appearing to be edging towards extra time, only for the midfielder to pop up in the 89th minute to secure his hat-trick and a memorable victory for Les Canaris.