Lazio were the better team in the first half but Juventus created the first opportunity when Paul Pogba headed wide.
Early in the second half Mandzukic missed a one-on-one opportunity while Pogba's effort was narrowly off target.
Manduzkic, who joined Juventus from Atletico Madrid in June, could not be denied on the hour when he rose to meet a perfect cross from Stefano Sturaro.
The Croatia forward created the second goal too, darting down the right flank before clipping a cross which Pogba brought down and Argentine Dybala crashed into the roof of the net to secure the team's seventh Super Cup title.
"It's my first game with Juventus, scoring and winning has made it an unforgettable one," said the 21-year-old striker, who joined Juventus in June from fellow Serie A side Palermo.
(Reporting by Amlan Chakraborty in New Delhi; editing by Martyn Herman)
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