He was drafted by the Colorado Rockies in the 38th round of the 2012 Major League Baseball Draft, but did not sign and attended Vanderbilt University. As a freshman, he played in only 11 games due to a broken bone in his foot and a torn labrum in his non-throwing shoulder. As a sophomore in 2014, he was named All-Southeastern Conference. After the Commodores beat the Virginia Cavaliers in the 2014 College World Series, he was named the College World Series Most Outstanding Player. He was a finalist for the 2015 Golden Spikes Award, presented annually to the nation’s top college player.

The Arizona Diamondbacks selected Swanson with the first overall selection of the 2015 Major League Baseball Draft. Swanson signed with the Diamondbacks, receiving a $6.5 million signing bonus. While batting in a simulated game against Yoan López in July 2015, Swanson was hit in the face with a pitch. The injury delayed his professional debut, which took place on August 12, 2015, with the Hillsboro Hops of the Class A-Short Season Northwest League. Swanson played in 22 games in his first professional season and finished with a .289 batting average, one home run, fourteen walks, and eleven RBIs.

On December 9, 2015, the Diamondbacks traded Swanson, Ender Inciarte, and Aaron Blair to the Atlanta Braves for Shelby Miller and Gabe Speier. The Braves invited Swanson to spring training, and he was sent to the Carolina Mudcats of the Class A-Advanced Carolina League at the end of spring training. Swanson was promoted to the Mississippi Braves of the Class AA Southern League on April 30, 2016, after hitting .333/.441/.526 in 21 appearances at Carolina. On June 28, he was named to appear in the All-Star Futures Game, soon after having played the Southern League All-Star Game. On August 16, the Braves announced that Swanson would be promoted to the major leagues the next day. Upon his callup, Swanson was assigned jersey #2. On August 17, he went 2-4 in his MLB debut, including his first career hit off of Minnesota Twins pitcher Kyle Gibson. Swanson hit an inside-the-park home run against the Washington Nationals on September 6, his team’s first such home run since 2001. With the hit, Swanson followed Paul Runge as only the second Braves player to ever make an inside-the-park home run with his first career big league home run. Swanson finished the year in the majors, recording a .302 batting average, .361 on-base percentage and a .803 on-base plus slugging percentage.

Prior to the start of the 2017 season, Swanson announced that he would again wear his college uniform number, 7. He had been assigned 2 because Gordon Beckham was wearing 7 at the time of Swanson’s call-up in August 2016. Beckham was later traded, leaving 7 available. Swanson struggled at the plate, recording a .185 batting average through May. Though he improved in June, Swanson had become the backup to Johan Camargo by the All-Star break. Swanson was subsequently sent down to the Gwinnett Braves on July 26, where he was expected to play every day. Swanson was recalled by the Braves on August 9 following an injury to Camargo.

Since December 2017, Swanson has been dating Mallory Pugh of Sky Blue FC and the United States women’s national soccer team. The couple met through Jace Peterson, who is Pugh’s brother-in-law and a former teammate of Swanson.

On May 4, 2018, Swanson was placed on the 10-day disabled list due to a left wrist strain. He returned to action on May 19, in a game against the Miami Marlins. Swanson again struggled throughout much of the season. From June 1 to August 21, 2018, he hit .229 with a .671 OPS. Swanson widened his batting stance in August, and began hitting with increased power. His offensive production lessened in September. On September 26, Swanson was injured during an at-bat against New York Mets pitcher Noah Syndergaard. The next day, he was diagnosed with a torn ligament in his left hand. Swanson was left off the National League Division Series roster due to his injury. The torn ligament led to soreness between his left index finger and wrist. As a result, Swanson underwent surgery on his left wrist in November 2018.

In 2019, Swanson had a productive first half, hitting .270 with 17 home runs before the All-Star break. However, he missed time due to a heel injury in the second half and struggled upon returning to the lineup, hitting only .204 with zero home runs over 142 second-half at-bats. For the season as a whole, he hit .251/.325/.422 with 17 home runs, 65 RBIs and 10 stolen bases.