Brewer attended Canton High School in Canton, Texas. The Pittsburgh Pirates selected Brewer in the fourth round of the 2011 MLB draft, and he signed. Brewer made his professional debut in 2012 with the Gulf Coast League Pirates, where he was 1–3 with a 3.24 ERA in eight games. He began 2013 with the Jamestown Jammers, but played in only three games due to injury, and subsequently missed all of the 2014 season. In 2015, Brewer pitched for the West Virginia Power where he was 5–9 with a 4.90 ERA in 22 starts. In 2016, he played with the Bradenton Marauders where he pitched to a 3–7 record with a 4.09 ERA in 18 games.
The New York Yankees selected Brewer from the Pirates in the minor league phase of the Rule 5 draft after the 2016 season, and he played within the Yankees farm system in 2017. In 41 relief appearances between three minor league teams, Brewer was 3–1 with a 2.82 ERA.
Brewer signed a major league contract with the Padres after the 2017 season. He began 2018 with the Triple-AEl Paso Chihuahuas. San Diego called up Brewer to the major leagues on April 12, 2018, and he made his debut that same night at AT&T Park against the San Francisco Giants. He was optioned back to El Paso the next day, and made additional appearances with the Padres during the latter half of the season. Overall, Brewer made 11 relief appearances for San Diego, recording a 1–0 record with 5.59 ERA and 10 strikeouts in innings.
Boston Red Sox
On November 20, 2018, the Padres traded Brewer to the Boston Red Sox for Esteban Quiroz. Brewer was included on Boston's Opening Day roster to start the 2019 season. On May 26, Brewer was optioned to the Triple-A Pawtucket Red Sox; to that point in the season, he had made 20 relief appearances, recording a 5.32 ERA and 0–2 record with 21 strikeouts in 22 innings. He was recalled to Boston on May 29, spending all of June and July with the Red Sox, until being optioned back to Pawtucket on August 4. Brewer was recalled to Boston on September 4, following the end of the Triple-A season. Overall with the 2019 Red Sox, Brewer appeared in 58 games, compiling a 1–2 record with 4.12 ERA and 52 strikeouts in innings. On March 26, 2020, the team optioned Brewer to the Double-APortland Sea Dogs.
