On February 15, 2008, the Florida Marlins signed Ozuna as an international free agent with a $49,000 bonus. He played for the Dominican Summer League (DSL) Marlins that year, where in 63 games, he hit .279 with six home runs and 43 runs batted in (RBIs) and eight stolen bases. In 2009, he played for the GCL Marlins, where in 55 games, he hit .313 with five home runs and 39 RBIs.

Ozuna began 2010 with the Greensboro Grasshoppers of the Class A South Atlantic League, but was sidelined after six appearances with an injured wrist. He returned in June with Short Season Jamestown, where he finished the season. In 74 total games, he hit .258 with 22 HR and 62 RBIs. Ozuna spent 2011 with Greensboro, where he hit .266 with 23 home runs, 71 RBIs, and 17 stolen bases over 131 games.

Ozuna was promoted to Jupiter Hammerheads of the Class A-Advanced Florida State League for the 2012 season, where in 129 games, he hit .266 with 24 home runs and 95 RBIs. That year, he led the Florida State League in home runs, RBIs, runs scored, and total bases. Ozuna was added to the Marlins’ 40-man roster on November 20, 2012. Baseball America rated Ozuna the 75th-best prospect in baseball prior to the 2013 season.

Ozuna opened 2013 with Jupiter, but after four games, he was promoted to the Jacksonville Suns of the Class AA Southern League. Ozuna was named the league’s player of the week on April 29, 2013. That same day, the Marlins promoted Ozuna to the major leagues following an injury to Giancarlo Stanton.

On April 30, 2013, Ozuna made his major league debut, and recorded his first career hit, a single off Jeremy Hefner of the New York Mets. He hit his first career home run (as well as his first RBI and run scored), a solo home run, off Cole Hamels of the Philadelphia Phillies, in his fifth game. Ozuna was used as the starting right fielder in Stanton’s absence. When Stanton returned, Ozuna became the starting center fielder. On July 22, Ozuna was optioned back to Jacksonville. Instead of joining Jacksonville, Ozuna was placed on the 15-day disabled list due to a ligament tear and avulsion fracture in his left thumb, which he injured while making a diving catch. On July 26, 2013, he underwent season-ending thumb surgery.

In 2014, Ozuna played a total of 153 games for the Marlins, batting .269 with 23 home runs and 85 RBIs. On September 11, 2014, he tied a franchise record with home runs in four consecutive games.

On July 5, 2015, Ozuna was sent down to AAA after going hitless in nine of ten games, dipping his average to .249 on the season. After playing 33 games and batting .333 with five home runs and 11 RBIs, he returned to the Marlins to complete the 2015 season, batting .278 with six home runs and 18 RBI.

In 2017, Ozuna was named NL Player of the Week for the week of April 10–16 after batting .435 with four home runs, 12 RBIs, and a 1.481 OPS.

Ozuna was named a starter for the NL in the 2017 MLB All-Star Game. Of the NL All-Star selections, he was one of eight to hit at least 35 home runs, but Ozuna was one of just four to walk at a rate of at least nine percent while striking out less than 22 percent. He established career highs in numerous categories, including batting .312 with 37 home runs and 124 RBI. He was named the NL Player of the Week (along with teammate Giancarlo Stanton once again for the week of September 18–24 after batting .500 (10–20) with three home runs, seven RBIs, and six runs scored.

On December 14, 2017, the Marlins traded Ozuna to the St. Louis Cardinals for Sandy Alcántara, Magneuris Sierra, Zac Gallen, and Daniel Castano. On January 12, 2018, it was announced that the Cardinals and Ozuna reached agreement on a one-year, $9 million contract. Prior to the 2018 season, Mike Petriello of MLB.com rated Ozuna the best left fielder in the major leagues, opining him as “a strong fielder,” and averaging the 12th-highest exit velocity [91 miles per hour (146 km/h)] over the prior three seasons of all hitters with at 1,000 plate appearances.

On June 2, 2018, Ozuna hit his 100th career home run off Chad Kuhl, in a 3–2 victory versus the Pittsburgh Pirates, and first home run at Busch Stadium as a member of the Cardinals. The following day, he hit a grand slam in a 5–0 win versus the Pirates.

He was named NL Player of the Week for the week of June 11–17, hitting .455/.478/1.000 (10-for-22, 1.478 OPS), with four home runs and eight RBIs in six games. This is his third Player of the Week award after winning it twice with the Miami Marlins during the 2017 season, in mid-April and early September. On July 30, 2018, at Busch Stadium versus the Colorado Rockies, Ozuna hit his first walk-off home run, a solo shot in the 10th inning, helping St. Louis defeat the Rockies 5–4. Ozuna finished his first season in St. Louis batting .280 with 23 home runs and 88 RBIs in 148 games.

Ozuna was placed on the injured list on June 29, 2019, with a finger fracture, and was activated on August 3. Over 130 regular season games and 485 at bats, Ozuna slashed .243/.330/.804 with 29 home runs and 89 RBIs.

On January 21, 2020, Ozuna signed a one-year contract worth $18 million with the Atlanta Braves.

In 2020 he batted .338/.431/.636 (each 3rd in the NL) with an NL-leading 18 home runs, 56 RBIs, and 145 total bases during the shortened 60-game season. Ozuna won his second Silver Slugger Award after the 2020 season; it was the first-ever Silver Slugger for designated hitter in the National League as a result of the universal DH rule implemented league-wide as a result of the COVID-19 pandemic.