Webb received a non-roster invitation to spring training with the Yankees in 2015, and opened the season with Scranton/Wilkes-Barre. Due to a tendon injury, he only pitched 38 innings during the season (going 2–3 with a 2.84 ERA), and the Yankees assigned Webb to the Arizona Fall League after the regular season.[7] In 2016, he pitched for Scranton/Wilkes-Barre,[8] going 4–3 with a 3.59 ERA over 72+2⁄3 innings pitched out of the bullpen.
On December 8, 2016, the Pittsburgh Pirates selected Webb from the Yankees in the Rule 5 draft.[9] He competed for a role with the Pirates during spring training,[10] but did not make the 25-man roster and was returned to the Yankees before the regular season.[11] Webb began the 2017 season with Scranton/Wilkes-Barre.[12]
The Yankees promoted Webb to the major leagues for the first time on June 22, 2017,[13] and he made his MLB debut on June 24.[14] He made seven relief appearances for the Yankees, pitching to a 4.50 ERA while striking out five and walking four in six innings.[15]
On April 14, 2018, Webb was claimed off waivers by the San Diego Padres.[18] He made one appearance for the El Paso Chihuahuas of the PCL before the Padres promoted him to the major leagues.[19]
St. Louis Cardinals
On June 29, 2018, Webb was claimed off waivers by the St. Louis Cardinals and assigned to the Triple–AMemphis Redbirds of the PCL.[20] He was recalled by St. Louis on July 27.[21] In 15+1⁄3 innings pitched for St. Louis, he compiled a 1.76 ERA.[22]
Webb began 2019 with Memphis. He was recalled by the Cardinals for the first time in 2019 on April 7,[23] and optioned back to Memphis on July 11.[24] He was recalled on July 21, finishing the season with St. Louis[25] Over 65 relief appearances during the regular season, he went 2–1 with a 3.76 ERA, striking out 48 over 55 innings.
In 2020 for the Cardinals, Webb recorded a 2.08 ERA and a 1–1 record in 21.2 innings pitched across 21 games.[26] After struggling to a 13.22 ERA in 22 appearances in 2021, Webb was designated for assignment by the Cardinals on June 3, 2021.[27] He was outrighted to the Triple-A Memphis Redbirds on June 6.[28] Webb elected free agency on October 14.[29]
Long Island Ducks
On April 4, 2022, Webb signed with the Long Island Ducks of the Atlantic League of Professional Baseball.[30] Webb made 53 appearances for the Ducks, working to a 2–5 record and 2.93 ERA with 68 strikeouts and 15 saves in 55+1⁄3 innings pitched.
Minnesota Twins
On February 8, 2023, Webb signed a minor league contract with the Minnesota Twins organization.[31] Webb was released by the organization on March 24.[32]