From 2009 through 2014, Weber pitched in the lower levels of Atlanta's farm system, reaching the Double-A level in 2014. Weber began the 2015 season with the Double-A Mississippi Braves. He was later promoted to Triple-AGwinnett, and began pitching games as a starter in late August. Weber was called up to the majors for the first time on September 8, 2015, to start in place of Matt Wisler, who had been forced to pitch in relief two days prior, and subsequently missed his turn in the rotation. Weber threw six innings against the Philadelphia Phillies, yielding two runs and four hits on 76 pitches. Overall with the 2015 Braves, Weber made five appearances, compiling an 0–3 record and 4.76 ERA. Weber was invited to spring training in 2016, and was sent to Gwinnett to start the season. At both the major and minor league levels, Weber pitched primarily in relief. He was recalled on April 18, after the team placed Gordon Beckham on the disabled list. Weber made three appearances and recorded a 10.57 ERA before returning to the minors on April 27. Jason Grilli was traded to the Toronto Blue Jays on May 31, and Weber was recalled again. Three days later, Weber was returned to Gwinnett. After a ten-day stint in the minors, Weber was recalled on June 13, only to be optioned the next day. Weber was recalled as an extra reliever and took struggling starter Matt Wisler's roster spot on July 31. On August 3, Weber was replaced with reliever Brandon Cunniff, only to be recalled on August 13. Overall with the 2016 Braves, Weber made 16 appearances, compiling a 1–1 record and 5.45 ERA.
Seattle Mariners
On November 2, 2016, Weber was claimed off waivers by the Seattle Mariners. He was designated for assignment by the Mariners on November 28, and outrighted to Triple-A on December 2. His contract was selected on May 13, 2017, so that he could make a start against the Toronto Blue Jays. Weber was placed on the disabled list the next day. Overall with the 2017 Mariners, Weber made a single appearance, allowing one run in innings pitched.
Tampa Bay Rays
On January 26, 2018, Weber signed a minor league contract with the Tampa Bay Rays. He was called up by the Rays on April 9, 2018, and was designated for assignment on April 13. He was called back up on May 13, designated for assignment again on July 14, and was outrighted on July 22. Overall with the 2018 Rays, Weber made two appearances, both in relief, allowing three earned runs in innings. Weber elected free agency on October 2, 2018.
Boston Red Sox
On December 21, 2018, Weber signed a minor league contract with the Boston Red Sox. ;2019 On May 6, 2019, the Red Sox called up Weber from the Triple-A Pawtucket Red Sox when David Price was placed on the injured list. Weber made his first appearance with Boston that day, pitching four innings of scoreless relief against the Baltimore Orioles. On May 23, Weber earned his first win as a starting pitcher, holding Toronto to one run on three hits in six innings. He was optioned back to Pawtucket on June 7, called up briefly during July and August, and recalled on September 1 when rosters expanded. Overall with the 2019 Red Sox, Weber appeared in 18 games, compiling a 2–4 record with 5.09 ERA and 29 strikeouts in innings. ;2020 Weber made Boston's opening day roster for the delayed start of the 2020 season, as a member of the starting rotation.
