Sampson was drafted by the Pittsburgh Pirates in the fifth round of the 2012 Major League Baseball draft and signed with the Pirates. He made his professional debut that season with the State College Spikes. He started nine of 11 games and had a 2.95 ERA. Sampson played the 2013 season with the Bradenton Marauders, finishing with a 5.14 ERA in 140 innings. He started the 2014 season with the Altoona Curve. In July he was selected to play in the Eastern League All-Star Game, but was forced to sit out the game by the Pirates. In August he was promoted to the Triple-A Indianapolis Indians after posting a 2.55 ERA with Altoona. Overall, he went 11–6 with a 2.96 ERA and 109 strikeouts.
Seattle Mariners
On July 31, 2015, Sampson was traded to the Seattle Mariners for J. A. Happ. Sampson made his Major League debut on June 18, 2016 against the Boston Red Sox. Sampson suffered a torn flexor tendon that required surgery in late June 2016 and was placed on the 60 day DL for the remainder of the season.
Texas Rangers
On November 2, 2016 he was claimed off waivers by the Texas Rangers. In 2017, Sampson rehabbed from the torn flexor tendon injury and made just 8 minor league appearances that season, going 2-2 with a 3.45 ERA in 31.1 innings with the AZL Rangers, Down East Wood Ducks, and the Round Rock Express. Sampson spent the first 5 months of the 2018 season with the AAA Round Rock Express, going 8-4 with a 3.77 ERA in 126.2 innings. Sampson was recalled to the Texas Rangers on September 4, and on September 11, 2018 Sampson made his first career MLB start. Sampson finished 2018 with a 0-3 record and a 4.30 ERA in 23 major league innings. Sampson was non-tendered by Texas after the 2018 season, and became a free agent on November 30, 2018. On December 27, 2018, Sampson re-signed with the Rangers on a minor league contract with an invite to major league spring training. Sampson was assigned to the AAA Nashville Sounds on March 28, 2019. On April 1, 2019, the Rangers purchased Sampson's contract and recalled him to the major league roster. On June 8, he recorded his first career complete game versus the Oakland Athletics. Sampson finished the 2019 season going 6–8 with a 5.89 ERA over 125.1 innings.
Lotte Giants
On November 22, 2019, Sampson signed one-year contract with the Lotte Giants of the KBO League.