Bergessio got his first after 13 minutes, meeting a pinpoint pass Lucas Castro before guiding home from a tight angle.
Siena's chances of a comeback were dealt a severe blow before the break as Dias Felipe was shown a straight red card for handling just outside the area, slipping at the crucial moment before landing on the ball.
Catania took complete control of the match with a one-man advantage, with Pablo Barrientos and Castan both going close to a second goal.
Bergessio eventually doubled the home side's advantage with a clever turn and finish, sending a measured low drive into the bottom corner.
The Argentine completed his hat-trick 20 minutes from time as he narrowly beat the offside trap to volley home from close range.