Webb was selected by the Yankees in the 10th round of the 2013 First-Year Player Draft out of the University of South Carolina, where he went 15-7 with 20 saves and a 2.34 ERA (173.0ip/45er) in a school-record 110 appearances over four seasons. He also helped the Gamecocks win the 2010 and 2011 College World Series titles while being named to the SEC Academic Honor Roll as a senior.
In 2014, Webb was named to the Eastern League's mid-season All-Star team while leading all Yankees farmhands (min. 60.0ip) with a 12.32 strikeouts per 9.0 innings pitched ratio while finishing second in New York's minor league system with 12 saves.
Since making his professional debut in 2013, Webb has posted a career record of 12-13 with 18 saves, a 3.48 ERA and 265 strikeouts in 129 appearances (five starts).