The Colorado Rockies selected him out of high school in the 38th round of the 2012 MLB Draft, but Swanson opted not to sign, instead playing college baseball for the Vanderbilt Commodores.

The Commodores won their first-ever national championship that season, and Swanson was named the 2014 College World Series Most Outstanding Player.

After one season in Arizona's farm system, he was traded to the Braves as part of a prospect package sent to Atlanta to acquire Shelby Miller.

Swanson made his MLB debut in August 2016 and was the only Atlanta rookie named to the Braves' 2017 Opening Day roster, but he struggled both offensively and defensively and was sent back to Triple-A that July.

[4] During his senior season on the Marietta Blue Devils basketball team, Swanson had a 44 percent three-point field goal shooting rate and averaged 14 points per game.

[6] Swanson had received recruitment offers from a number of universities, including Troy, Clemson, and Georgia Tech, but he agreed to attend Vanderbilt after a conversation with coach Tim Corbin.

[9] Swanson finished the regular Southeastern Conference (SEC) season with a team-leading .366 batting average, 47 runs scored, 21 doubles, and 17 stolen bases, and he was an All-SEC First Team selection.

[13] Vanderbilt's postseason run concluded with their first ever national championship when they defeated Virginia 3–2 in the College World Series (CWS) finals.

[19] After finishing the regular season with a .347 batting average, 50 RBI, 34 extra-base hits, and a conference-leading 60 runs scored,[20] Swanson was both named to the All-SEC Second Team and was a semifinalist for the Dick Howser Trophy, given to the best college baseball player in the United States.

[31] His professional baseball debut was delayed when, during a simulated game before assignment to one of the Diamondbacks' minor league affiliates, Swanson was hit in the face by a fastball from pitching prospect Yoan López.

[33] Having missed six weeks with the concussion, Swanson was told that he would finish out the season with Hillsboro rather than receiving a late-season promotion that would push his development.

[35][36] On December 9, 2015, the Diamondbacks traded Swanson, Ender Inciarte, and Aaron Blair to the Atlanta Braves in exchange for Shelby Miller and Gabe Speier.

The Braves were in the middle of a rebuild, and president of baseball operations John Hart said that the team "wanted to make it painful for [Arizona] with players that we got back" by refusing a Miller trade until they received significant prospects like Swanson.

[38] He played in 22 games there, batting .333 with a Carolina League-leading 12 doubles, before receiving a promotion to the Double-A Mississippi Braves at the end of April.

[35] Swanson was called up to Atlanta on August 16, 2016, after the Braves traded shortstop Erick Aybar to the Detroit Tigers to clear the position for him.

[46] He remained with the Braves through the remainder of their 2016 season,[47] finishing the year with a .302 batting average, three home runs, 17 RBI, and 11 extra-base hits.

[48] Swanson was the only rookie to make the Braves' 2017 Opening Day roster, joining Chase d'Arnaud, Freddie Freeman, Adonis Garcia, Jace Peterson, and Brandon Phillips in the infield.

[49] He began the season in a sophomore slump both offensively and defensively: by the end of May, Swanson was batting .185 with a .559 on-base plus slugging (OPS), while he had also committed 11 errors at shortstop.

[59] He began feeling discomfort in his left wrist at the end of April, and after aggravating the injury during a game against the New York Mets, he was placed on the 10-day disabled list on May 4.

[61] He again struggled through most of the season, batting .234 with a .704 OPS between June 1 and August 21, a stretch that ended with Swanson's first career multi-home run game.

[66] The Braves lost to the Dodgers in the four game NLDS,[67] while Swanson's hand and wrist continued to bother him until November 5, when he underwent surgery to remove a loose piece of cartilage.

[71] With his wrist no longer bothering him, Swanson showed a new offensive power during 2019 spring training, giving manager Brian Snitker hope that his production would increase.

[74] On July 23, however, Swanson landed awkwardly on first base while attempting to run out a ground ball in a game against the Kansas City Royals, injuring his left heel in the process.

Adam Duvall followed this with a two-RBI single, and Mark Melancon held the Cardinals scoreless in the bottom of the ninth to win the game 3–1 for Atlanta.

The Braves began their abbreviated season with a three-game series against the New York Mets, during which Swanson went 5-for-12 at the plate with a double, a home run, and six RBI.

[88] The Braves lost the 2020 NLCS to the Los Angeles Dodgers in seven games,[89] and Swanson cost the team a potential scoring opportunity on a fourth-inning baserunning error.

He broke for home plate on a ground ball from Nick Markakis and was caught in a rundown by third baseman Justin Turner and catcher Will Smith.

[98] Swanson's defensive abilities were on display against the Brewers in the 2021 NLDS: in Game 3, he held Milwaukee scoreless on separate occasions, first by preventing runners from advancing on a groundout from Lorenzo Cain and then by turning an eighth-inning double play to retire Jace Peterson and Willy Adames.

[35] The Braves faced the Dodgers in the 2021 NLCS, where in Game 1 Swanson tagged out Chris Taylor to prevent Los Angeles from breaking a 2–2 tie in the top of the ninth inning.

In the seventh inning, however, he hit a 95 mph (153 km/h) fastball from Houston Astros reliever Cristian Javier for a game-tying home run.