Milinkovic-Savic's fine header set Lazio on their way with eight minutes left at the Stadio Olimpico, seconds after coming on as a substitute, before Correa sealed victory in the closing moments of the match.
The victory books Lazio a ticket to next season's Europa League, much-needed for Simone Inzaghi's side as they are eighth in Serie A, four points off the European berths with two games left to play.
"We really wanted this Cup," said Inzaghi.
"It was a beautiful evening in our stadium with our fans.
"It's a great satisfaction, we deserve it because we've come a long way."
His side's win was marred by clashes outside the stadium between Lazio's hardcore "Ultras" supporters and police before the game.
Fans threw projectiles at officers as a police car was burned near the Ponte Milvio bridge not far from the Stadio Olimpico.
According to Italian media, two police officers were injured, one in the ear by a projectile, and five fans were arrested.
Atalanta had eliminated holders Juventus in the quarter-finals and been bidding to win their second Coppa Italia and their first major honour since they won the competition in 1963.
For the side from Bergamo -- who are still in the running to qualify for the Champions League for the first time -- it is their third defeat in the final after missing out in 1987 and 1996.
Gian Piero Gasperini's side head to Serie A champions Juve on Sunday looking to keep hold of fourth place, with AC Milan and Roma both three points back.
"It's a pity, but finals are like that," said Gasperini. "The decisive moment of the match arrived in the second half and we lost on a corner kick.
"We now head to Sunday, because there is a place in the next Champions League to defend. We'll overcome this disappointment, even if our run in the tournament was a great experience."
Atalanta had arrived in Rome on a 12-match unbeaten run since the end of February. They had beaten Lazio in both Serie A meetings this season.
But they could not find a way past their solid opponents and failed to take their chances in a nervy game, with Papu Gomez denied by Thomas Strakosha in the Lazio goal within the first minute.
Atalanta's top scorer Duvan Zapata and midfielder Marten De Roon also squandered two quickfire efforts on 26 minutes, before Gomez hit the post with 14 minutes left.
Seconds afterwards Milinkovic-Savic made an almost immediate impact after coming on after a month out injured to replace Luis Alberto, who had just missed a chance for Lazio.
The Serb broke through leaving Pierluigi Gollini powerless, and Correa finished off a counter-attack minutes later to award Inzaghi his first major honour since winning the competition as a Lazio player in 2009.