Eaton played college baseball for Miami University before the Diamondbacks selected him in the 19th round of the 2010 MLB draft. He made his MLB debut in 2012 with the Diamondbacks, and played with them through the 2013 season, when they traded him to the White Sox. He played for the White Sox from 2014 to 2016 and the Nationals from 2017 to 2020. He was also a member of the Nationals' 2019 World Series champion team.