Premier League leaders City were fortunate to go ahead after 57 minutes when Vincent Kompany, climbing all over Giorgio Chiellini, saw the defender head an own goal as the home skipper wheeled away to celebrate instead of being punished for a foul.
But Juve pulled level in the 70th minute when Mandzukic touched the ball in off the post after midfielder Paul Pogba sent in a high cross from the left.
It was the first goal City had conceded after five domestic league wins so far this term but worse followed when Morata scored with a 20-metre left-foot curler in the 81st minute.
City manager Manuel Pellegrini thought his team had played the better football, telling BT Sport: "It's disappointing, I don't think we deserved to lose. We had better chances and played better than Juventus.
"It was a typical Champions League game, both teams didn't create many chances but we had more possession. It was not a tactical problem, Juventus just scored two beautiful goals," he said after City failed to win their first home match for a fifth successive Champions League campaign.
They had a great chance to take a grip on the game in the opening minutes when Yaya Toure set up Raheem Sterling but the winger wasted the chance when his weak shot was saved by goalkeeper Gianluigi Buffon's feet.
City's Wilfried Bony, in for striker Sergio Aguero, who managed to come on for the last seven minutes despite nursing a knee problem, should have done better midway through the first half when he got clear of the defence before slicing wide.
Fernandinho then went close for the hosts with a fierce shot that flew over the bar but Juve otherwise looked comfortable, although they did not seriously test City keeper Joe Hart until Stefano Sturaro forced him into a smart save after an hour.
However last season's beaten Champions League finalists, without a win in their three opening Serie A matches, found the net twice in the closing stages to snatch all three points.
(Writing by Mike Collett; Editing by Ken Ferris)
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