Things looked to be going to plan for Antonio Conte's side as they took a 30th-minute lead through Mirko Vucinic at the club's new 41,000-seater stadium.
However, Vucinic then saw red after picking up his second caution on the stroke of half-time to leave Juve with 10 players, and Bologna took full advantage to equalise through Daniele Portanova at the start of the second half to claim their first point after two defeats.
Juve, who recorded wins over Parma and Siena in their opening two games, had the first chance of the match but Milos Krasic lost his footing before he could shoot after being found with a great pass from Vucinic.
Bologna responded with three efforts of their own in a 12-minute spell, Panagiotis Kone bringing a save out of Gianluigi Buffon, Gaston Ramirez firing over from distance and Kone spurning a good opening when he shot wide following a rapid counter-attack.
However, Bologna's hopes of causing an upset suffered a blow on the half-hour mark when they fell behind. Andrea Pirlo's quickly-taken free-kick picked out Vucinic and he scored from close range.
That goal sparked a good spell for the home side and Alessandro Del Piero fired wide before Krasic drew a superb save out of Jean Francois Gillet.
Juve could not add a second goal before the break, though, and they went into the interval a man down after Vucinic was booked for a second time following a needless foul.
It took Bologna just seven minutes of the second half to make their numerical advantage count with Portanova climbing above Giorgio Chiellini at a corner to head in the equaliser.
Despite that setback and playing with a man less, however, it was Juve who looked the more likely winners as they created a number of chances in the final half-hour.
Pirlo's curling effort looked to be heading in until Gillet intervened, and the visiting goalkeeper also kept out Andrea Barzagli's header before tipping over Simone Pepe's cross-shot.
Bologna threatened a late winner when Marco Di Vaio shot wide in the 85th minute before the final opportunity fell Juve's way in the 88th minute, but Alessandro Matri failed to hit the target after turning well inside the area.