[4] As a freshman at Vanderbilt in 2015, Wright appeared in 29 games with three starts and went 6–1 with a 1.23 ERA, 62 strikeouts and four saves.

[15] Wright was promoted to the Florida Fire Frogs in August 2017, and finished the season there.

[29] He made his postseason debut in October against the Miami Marlins during the National League Division Series, pitching six innings of shutout ball in the Braves' Game 3 and series-clinching victory.

[30] He lasted less than an inning in his sole National League Championship Series start, allowing seven earned runs in Game 3 to the Los Angeles Dodgers.

[32] On October 26, 2021, it was announced that Wright was added to the Braves roster for the World Series against the Houston Astros.

Wright proceeded to work out of the jam with only one inherited runner scoring, then pitched four subsequent innings, allowing only one additional run.

[39] On September 24, Wright won his twentieth game of the season, and became the first Braves pitcher since Russ Ortiz in 2003 to reach that milestone.

[42] Wright matched Ortiz's 2003 win total on October 1, 2022, and ended the regular season as the major league leader in victories.

A Braves pitcher had not led MLB in wins since Tom Glavine's 2000 season.

He made seven starts and nine total appearances for Atlanta, working to a 1–3 record 6.97 ERA with in 31 innings pitched.

[50][51] On November 17, 2023, the Braves traded Wright to the Kansas City Royals in exchange for pitcher Jackson Kowar.