Fried (9-2) combined with three relievers on a seven-hitter. He gave up only five hits, including two to Albert Pujols, with one walk and four strikeouts in winning his nine consecutive decision after losses in his first two starts.
Fried was pulled after experiencing tightness in the back of his upper right leg.
''At that point of the game we wanted to be as careful as we could,'' Fried said.
Manager Brian Snitker said he ''didn't want to take chances'' with leaving Fried in the game, but said he expects the left-hander to make his next start.
Fried and Miles Mikolas (5-7) each threw four scoreless innings before Ozuna led off the fifth with his 17th homer deep into the left-field seats. Rosario followed with his first homer.
Rosario returned from the injured list on Monday after needing laser surgery to correct blurred vision and a swollen right retina, forcing him to miss 62 games.
''I was nervous,'' Rosario said through a translator. ''I wasn't sure I could go out there and be myself and play the way I knew I can play. I'm glad to have the first homer.''
Austin Riley and Travis d'Arnaud added back-to-back doubles off Mikolas in the sixth, giving Atlanta a 3-0 lead.
St. Louis loaded the bases against right-hander Jesse Chavez in the eighth, but pinch-hitter Nolan Gorman grounded out to end the threat.
The Braves remained 2 1/2 games behind the New York Mets in the NL East. Atlanta is 26-7 since June 1, when it was 10 1/2 games behind the Mets.
The Braves flexed their power game with the homers by Ozuna and Rosario.
Atlanta has seven players, including two catchers, with 10 or more homers before the All-Star break for the first time since the franchise moved to Atlanta in 1966. The Braves lead the NL and rank second in the majors with 126 homers. The New York Yankees began the night with 133.
''That's kind of how our team is,'' Snitker said. ''... These are big, strong guys who are hitting it.''
Left-hander A.J. Minter, who has shared the closer's role with Will Smith while Kenley Jansen is on the injury list with an irregular heartbeat, pitched the ninth for his second save.
The Cardinals have lost four straight, their longest skid of the season, including the first three of the four-game series against Atlanta. They remain three games behind division-leading Milwaukee in the NL Central.
Mikolas allowed three runs on five hits in six innings, ending his streak of allowing no more than two earned runs in five consecutive starts and bemoaned the mistakes to Ozuna and Rosario.
''It's tough, two pitches kind of got away from me,'' Mikolas said. ''... Sometimes you make mistakes and they hit singles, and sometimes they hit home runs.''
The Cardinals left nine runners on base and have been outscored 16-4 in the first three games of the series. Tommy Edman had one hit as he returned to the leadoff spot for St. Louis after hitting seventh on Tuesday night.
''I know the hitters will come around,'' Mikolas said. ''Sometimes you have to bide your time. They will show up, I'm sure.''
Riley leads the Braves with 21 homers, followed by Ozuna (17), Dansby Swanson (14), Matt Olson (12), catchers Travis d'Arnaud (11) and William Contreras (10) and Adam Duvall (10).
Cardinals: LHP Steven Matz (left shoulder) is scheduled to throw 40 to 60 pitches for Triple-A Memphis on Thursday. Manager Oliver Marmol said Matz likely will need another rehab start before he comes off the injury list and joins the Cardinals' rotation. ''Do we have a need? Yes,'' Marmol said. ''Do we want to rush because we have a need? No.''
Braves: OF Adam Duvall is expected to start on Thursday for the first time since he was hit by a pitch on his left hand Saturday at Cincinnati. He played off the bench for the second straight game. ... RHP Silvino Bracho was outrighted to Triple-A Gwinnett after being designated for assignment on Monday.
The Braves will try to complete a four-game sweep when RHP Spencer Strider (4-2, 2.87 ERA) faces Cardinals LHP Matthew Liberatore (2-1, 5.66) in a matchup of rookies on Thursday night.
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