[2] Webb made his professional debut in 2014 with the Rookie-level Gulf Coast Braves, going 2–1 with a 2.14 earned run average (ERA) in 33+2⁄3 innings pitched.

Webb split the 2017 season between the High–A Florida Fire Frogs and the Double-A Mississippi Braves, accumulating a 5–2 record with a 2.07 ERA in 65 innings.

[7][8] He made his major league debut on April 16, recording one-third of an inning pitched via a strikeout of Ketel Marte.

[22] Webb did not make an appearance with the Diamondbacks, spending his entire tenure with Arizona's Triple-A affiliate, the Reno Aces.

[24] On June 7, 2022, Webb was traded to the Atlanta Braves in exchange for cash considerations.

[27] In 28 appearances for Gwinnett, he logged a 2-3 record and 5.34 ERA with 35 strikeouts and 3 saves across 30+1⁄3 innings pitched.

[29] He was assigned to the Triple-A Salt Lake Bees to begin the 2023 season, where he made 16 appearances and struggled to a 6.75 ERA with 21 strikeouts in 17+1⁄3 innings pitched.

[30] Two days later, he made his first pitching appearance since 2021, throwing a scoreless inning against the Boston Red Sox.

Webb made 60 relief outings for Baltimore in 2024, compiling a 2–5 record and 3.02 ERA with 58 strikeouts across 56+2⁄3 innings pitched.