The Montenegro striker, signed from AS Roma for 15 million euros in the close-season, turned in a free-kick on the half-hour as Bologna were caught off-guard only to be dismissed for two senseless yellow cards just before halftime.
Daniele Portanova levelled for the visitors after the break.
Champions AC Milan remained winless as they were held 1-1 at home by Udinese with Antonio Di Natale, last season's league top scorer, netting for the third straight game for the visitors.
Napoli, who beat Milan 3-1 on Sunday, came down to earth with a bump as they lost 1-0 at Chievo, Davide Moscardelli scoring the winner midway through the second half.
Germany striker Miroslav Klose notched the winner for Lazio as they secured a 2-1 victory on Cesena's artificial pitch and Atalanta won by the same score at Lecce to remain unbeaten.
Promoted Atalanta moved up to one point, having started the season with minus six as a penalty for their involvement in a match-fixing scandal in Serie B.
Argentine striker German Denis scored both their goals, including a penalty, taking his tally to three.
Cagliari, who won their first two games along with Juventus, Napoli and Udinese, were beaten 3-2 at Palermo.
Juventus, Udinese and Genoa are level on seven points from three games at the top, Genoa beating visitors Catania 3-0 with two goals from Argentine striker Rodrigo Palacio.
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Juventus coach Antonio Conte said: "It's disappointing not to win but I'm satisfied for the effort today and in general for the three matches so far.
"We know where we have to improve, especially in the matter of self-belief after two seasons which were not easy," he added on the club's website.
Juventus, a disappointing seventh last season, made a confident start and deservedly went ahead in the 27th minute.
Andrea Pirlo took a quick free-kick in midfield while the Bologna players still had their backs turned and with the ball still moving.
His pass found Vucinic unmarked in the penalty area and he turned the ball past Jean-Francois Gillet with the visitors wondering what had happened.
Vucinic had already been booked for dissent and could have been sent off for pulling a cardboard box over his head in the celebrations.
But there was no let-off in first half stoppage-time when he caught Archimede Morleo with a late tackle deep in the opposition half and was dismissed amidst angry Juve protests.
Bologna levelled seven minutes after the break when Portanova rose above the defence at a corner to head past Gianluigi Buffon, who stayed rooted to his line.
Pirlo and Giorgio Chiellini had chances to put Juventus back in front but Gillet kept the hosts at bay.
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