Vucinic's 72nd minute penalty broke the resistance of Pescara, who had Giuseppe Rizzo sent off for the foul on Arturo Vidal that led to the spot-kick, before the Montenegrin doubled their lead five minutes later with a smart curling strike.
Bottom side Pescara pulled a goal back through Emmanuel Cascione's long-range drive seven minutes from time.
Antonio Conte's side, who are 2-0 down heading into Wednesday's Champions league quarter-final, second leg with Bayern Munich in Turin, have 71 points from 31 games.
Pescara, who have only 21 points and look certain to be relegated, had keeper Ivan Pelizzoli to thank for keeping the score down after he pulled off a series of magnificent saves.
"I'm happy with the display, even if in the last few minutes after Pescara's goal we should have been calmer," Conte told Sky Sport Italia.
"It wasn't easy today because we were up against a side that had to defend, and it seemed as though we were cursed in front of goal.
"I'm relieved. Those three points were fundamental," he added.
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The champions laid siege to the Pescara goal in a dominant performance where they created plenty of good chances.
Pelizzoli's first serious save came after 12 minutes when Stefan Lichsteiner fed a cross from the right towards Emanuele Giaccherini, who span and hit a powerful volley that the keeper did well to tip away.
Five minutes later, Sebastian Giovinco shot from close range straight at the onrushing Pelizzoli. The 32-year-old keeper also somehow tipped Fabio Quagliarella's low, powerful drive on to his right-hand post six minutes before the break.
Vucinic was the worst culprit in a wasteful Juventus attack, first mis-hitting the ball with an open goal at his mercy on 50 minutes after Kwadwo Asamoah's shot was pushed out to him by Pelizzoli, then shooting from an almost impossible angle in the area when Giaccherini was standing unmarked on the edge of the six-yard box.
Juve finally took the lead with 17 minutes left when Rizzo pulled back Vidal as he went to meet Vucinic's through ball and referee Sebastiano Peruzzo gave a penalty.
Vucinic appeared to have ended the match as a contest when he curled home a smart finish five minutes later, redeeming himself for his shocking earlier miss.
An injury time tap-in from Rolando Bianchi gave Torino a 2-2 draw at Bologna in their mid-table clash.
Torino took the lead through Paulo Barreto after 25 minutes, but were pegged back by a powerful header from Panagiotis Kone just after an hour and a fine Tibero Guarente strike five minutes from time.
However, Bianchi stepped up to bag a point for his side, who have 36 points but stay in 14th place, while Bologna remain in 12th on 37.
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