Ozuna belted a two-run homer that gave the Marlins a three-run cushion in the fifth inning, and Yelich matched a career high with four hits, including two doubles.
Ozuna and Yelich each scored two runs.
"It's just one of those days where the balls found some holes," Yelich said. "I put some pretty good swings on some balls. That's how it works sometimes. It was a good win for us. We did a good job battling back after we got down, 3-2, there."
Miami watched an early two-run lead disappear when the Braves scored three times in the third inning. Miami had pulled ahead by two in the second inning, with Ozuna getting the rally going with a leadoff walk.
In the fifth, the Marlins went ahead for good, scoring three times, and capitalizing on a Nick Swisher error in right field.
Yelich's single to right put runners on the corners. The Marlins went ahead, 4-3, on Martin Prado's sacrifice fly to right field. On the play, Swisher dropped the ball, allowing Prado to reach. With two outs and one on, Ozuna gave Miami some breathing room with his two-run homer to left.
The shot was his eighth of the season and second of the road trip, with the other last weekend at Washington.
"I think overall, Ozuna is starting to get his swing back," manager Dan Jennings said. "His rhythm is starting to be there. Definitely his balance is much better. That was big. We were able to add on some runs down the stretch."
Having Yelich in left field and Ozuna in center is a welcome sight for Miami. The team opened the season with what they thought was the best young outfield in the National League.
But getting it on the field at the same time has been a challenge. Stanton, who hasn't played since June 26, aggravated his left hand on Tuesday in a rehab assignment game. His status remains uncertain.
Yelich is dealing with his own issue, a bruised right knee.
"Nobody is at 100 percent right now," Yelich said. "It's that time of the year. You've got to battle through some stuff. I feel fine. Good enough to play."
Joe Frisaro is a reporter for MLB.com. He writes a blog, called The Fish Pond. Follow him on Twitter @JoeFrisaro and listen to his podcast. This story was not subject to the approval of Major League Baseball or its clubs.