The Old Lady picked up the lead in the first half via Mirko Vucinic, but it was unable to collect the three points as Daniele Portanoca equalized after Juve went down to 10 men.
Mirko Vucinic opened the scoring around the half-hour mark, but was dismissed in the closing stages of the first half, and Daniele Portanova then restored parity after the interval.
Juventus dominated proceedings right from the start and had a number of chances to open the scoring as Milos Krasic and Vucinic in particular looked lively early on. The Montenegro international tried his luck with a shot from 20 yards out after only seven minutes, but failed to trouble Jean-Francois Gillet.
Vucinic had more luck, though, as he escaped the attention of the Bologna defense after a free-kick from Andrea Pirlo and had no trouble breaking the deadlock with a low shot.
The home side was dealt a huge blow in the closing stages of the first half when it was reduced to ten men. Vucinic went from hero to zero as he fouled Archimede Morleo inside the Bologna half and was consequently shown his second booking of the evening.
On 52 minutes, Bologna restored parity after a set piece. Gaston Ramirez found Daniele Portanova from a corner kick and the defender headed home to level the score.
Conte's men immediately took back the initiative and they could easily have grabbed the lead again. Both Pirlo and Giorgio Chiellini came close to adding their name to the scoresheet on the hour mark before Andrea Barzagli squandered two big chances to edge his team back ahead, and in the end the Turin side had to settle for a single point.
Juve resumes match action on Sunday when it meets Catania at Sicily, while Bologna's next match is the home game against Inter on Saturday evening.