The Croatia forward struck in the 18th minute, just after Fernandinho had missed a golden opportunity for the visitors who had already made sure of their place in the knockout stage.
In a match which only sporadically burst into life, Stefano Sturaro hit the post for Juventus and Raheem Sterling squandered another excellent chance for City 10 minutes from time.
Juventus completed a double over the English Premier League side, who had already qualified from Group D after winning their previous three games.
The Serie A champions, still unbeaten after five games, now top the group with 11 points and visit Sevilla in their final match.
Juve coach Massimiliano Allegri, who has frustrated supporters by frequently leaving out forward Paulo Dybala, picked the Argentine alongside Mandzukic in attack, although fans would have rather seen him partner Alvaro Morata.
"I made the decision to field Mandzukic a couple of days ago, we needed a player with his characteristics," Allegri said in a touchline television interview. "This is the most difficult group in the Champions and the lads have done well."
Mandzukic answered his critics in the 18th minute when Alex Sandro sent over a cross from the left and the Croatia forward held off Nicolas Otamendi's challenge to volley the ball in from close range.
The goal came just after Fernandinho had fired over from close range when he had the whole Juventus goal to aim at.
With Paul Pogba in inspired form in midfield for Juventus, Mandzukic then forced a good save from Joe Hart who also blocked an effort from Stephan Lichtsteiner after the Swiss had burst down the right.
Morata came on for Mandzukic in the 54th minute and almost immediately set up a second.
The Spaniard burst clear and lifted his effort over Joe Hart but wide of the goal and Sturaro hit the post from point-blank range as he tried to prod it in.
Hart made another good save to deny Dybala after he got free, then Sterling failed to score from close range with Juve goalkeeper Gianluigi Buffon caught off balance.
Sevilla, beaten 4-2 at Borussia Moenchengladbach, must beat Juventus in their last game and hope the Germans lose to City so they finish third in the group and go into the Europa League.
(Writing by Brian Homewood in Berne; Editing by Rex Gowar)
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