Juve, who beat Lazio 2-0 in the Italian Cup final at the same venue in May after they had also won the league title, came back from a two-goal deficit in a frantic five-minute spell before Murgia netted the winner.
Lazio dominated the opening hour as striker Ciro Immobile fired them ahead with a 32nd-minute penalty after he was brought down by Juventus keeper Gianluigi Buffon and then headed a delightful Marco Parolo cross into the far corner in the 54th.
Les joueurs de la Lazio lors de la Supercoupe d'Italie.
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Looking out for the count, Juventus staged a late comeback as Argentina forward Paulo Dybala put them on level terms with goals in the 85th and 90th minutes.
Dybala curled in a superb free kick from 25 metres and then converted a penalty, throwing vociferous Juventus fans into raptures as both lots of supporters created a cracking atmosphere on the terraces.
But the Bianconeri's celebrations were short-lived as Murgia swept in a low cross from the left at the death, delighting the Lazio contingent as he settled a rip-roaring curtain raiser to the Italian season.
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