The hosts were handed the initiative in the second half when Marco Motta, on loan from Juventus, was sent off for a second booking with the score 1-1.
Juve fell behind after only four minutes when Pablo Barrientos turned on the edge of the area and finished with a pinpoint left-footed effort.
Catania's lead lasted only 18 minutes as Andrea Pirlo left goalkeeper Tomas Kosicky rooted to the spot when he curled a free-kick around the wall and into the net.
The match turned in the 65th minute when Motta was dismissed for a clumsy tackle from behind on Paolo De Ceglie.
Chiellini then rose to head into an unguarded goal after Kosicky missed a free-kick from Pirlo with 16 minutes remaining and Quagliarella rounded off the night with an effort that deflected into the net off the goalkeeper seven minutes later.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.