Having replaced Sebastian Giovinco in the second half, Vucinic received a pass from Paolo De Ceglie and slotted a shot in from the edge of the box as Milan's defense struggled to keep up.
Milan had gone ahead six minutes in at Juventus Stadium when Stephen El Shaarawy scored with a bouncing shot from the edge of the area. Giovinco equalized for Juventus in the 12th minutue from a free kick.
The victory was a big relief for Juventus following a painful 2-1 loss at home to 10-man Sampdoria on Sunday.
''This was a great response,'' Juventus coach Antonio Conte said.
Juventus and Lazio are currently first and second in Serie A, with the Turin club five points ahead.
Juventus lost its only match last season to Napoli in the Italian Cup final.
This season's final is scheduled for May 26.
Milan was fortunate not to be reduced to 10 men in the 41st when defender Philippe Mexes made contact with Juventus goalkeeper Marco Storari after having already picked up a yellow card. But the referee didn't award a foul.
Milan had an excellent chance to equalize in the 119th and send the match to a penalty shootout, but Bakaye Traore volleyed directly at Storari, then El Shaarawy missed a minute later.
''We had chances to equalize and we weren't able to, but we played a solid match,'' Milan coach Massimiliano Allegri said.