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Atalanta had been the better side at Lazio's Stadio Olimpico home, hitting the woodwork either side of half-time in the showpiece match.
But substitute Milinkovic-Savic broke the deadlock for Lazio, powering home Lucas Leiva's corner, before Joaquin Correa wrapped it up with a second goal in the 90th minute.
Lazio's first Coppa Italia crown since 2013 will go some way towards making up for losing two of the past four finals to Juventus.