The Marlins were trailing 1-0 in the sixth when they mounted a rally aided by the Twins miscue.
That brought up Ozuna, who lined a single up the middle to score Lucas with the tying run. Center fielder Clete Thomas then mishandled the ball, allowing Stanton to trot home for a 3-2 lead.
Ozuna later added a run-scoring double and Derek Dietrich homered for Miami, which took a win in the first of a two-game interleague set.
The rally made a winner of A.J. Ramos (2-2) after Miami starter Jose Fernandez had held the Twins to one run on four hits over five innings.
Steve Cishek recorded a 1-2-3 ninth for his 13th save, securing the victory.
Correia (6-5) absorbed the loss, surrendering two runs on five hits over six full frames.
Josh Willingham and Trevor Plouffe each drove in a run for Minnesota.