Kucher deflected a cross intended for Juve's Sebastian Giovinco into his own net in the 56th minute to give the Italians the win at the Donbass Arena.
Juventus midfielder Kwadwo Asamoah nearly got an own goal himself later in the second half as he turned to deal with a Razvan Rat cross, but the ball struck the post after coming off his thigh.
The victory gave Juventus 12 points and took them through with Shakhtar, who had held the Italian side to a 1-1 draw in Turin in October.
Holders Chelsea finished third, despite having the same points as the Ukrainians, thanks to a poorer head-to-head tally, and were sent off to the Europa League.
Andrea Pirlo had his claims for a penalty ignored following an apparent handball by Shakhtar defender Fernandinho in the penalty area before Juventus missed a chance to go ahead when Giovinco, alone against the goalkeeper, narrowly missed the target on 29 minutes.
Mirko Vucinic shrugged off his marker and passed to the unmarked Giovinco who shot just wide.
Pirlo hit the far post from the edge of the area before Juventus's constant pressure paid off with the goal as Kucher tried in vain to deflect a low cross from the right by Stephan Lichtsteiner.
Juventus defender Leonardo Bonucci said the players had worked hard to get into the last 16 for the sake of coach Antonio Conte, who is nearing the end of a four-month ban for involvement in a match-fixing scandal.
"We've paid him back by getting into the last 16 - we owed it to him," Bonucci told www.juventus.com. "He's suffered being away from the pitch these last few months and now he'll come back stronger than ever."
"Juventus reached their goal and progressed into the next round despite the absence of our coach on the touchline," assistant coach Angelo Alessio told a news conference.
"We are quite optimistic about our future in the Champions League."
Shakhtar coach Mircea Lucescu was disappointed with his team, saying on television: "It was motivation that mattered in this game. Juventus were more motivated and aggressive as they wanted to make their way into the playoffs and it affected the result.
"We lost too many clashes in midfield and in this situation the suspended Luiz Adriano would have been a great help as he is always tackling for the ball in games like this."
The Brazilian forward was banned for one game for unsporting behaviour during last month's match against Nordsjaelland when he scored as his team tried to play the ball back to their opponents' goalkeeper following an injury to a player on the Danish side.
