Webb's departure comes after he served as the backup to Eli Manning during his rookie season. Fourth-round rookie Kyle Lauletta and fellow quarterback Alex Tanney remain on the roster for the Giants, who have the 37-year-old Manning under contract through the 2019 season.
The 23-year-old Webb was selected by the Giants with a third-round pick of the 2017 NFL Draft. New York elected against taking a quarterback in the first round of the 2018 NFL Draft as they chose former Penn State running back Saquon Barkley with the second overall pick.
Webb, who is listed at 6-foot-5, did not record a snap in a regular-season game in 2017. He finished the preseason by completing 28 of 53 passes for 283 yards with one touchdown and no interceptions.
The Giants claimed defensive end Mario Edwards among their six selections on Sunday. Edwards, who was a former second-round pick of Oakland, was waived by the Raiders on Saturday as the team trimmed its roster to 53 players.
Oakland unsuccessfully attempted to shop the 24-year-old Edwards prior to the roster deadline, NFL Network's Tom Pelissero reported.
Edwards recorded 71 tackles and 5.5 sacks during three seasons with the Raiders. He missed nearly all of the 2016 campaign with a hip injury.
The 6-3, 280-pound Edwards collected a career-high 3.5 sacks to go along with 27 tackles while starting in all of his 14 appearances last season.
A new beginning with the Giants should be beneficial for Edwards, who will attempt to push rookie B.J. Hill and second-year player Dalvin Tomlinson.
Wide receiver Kaelin Clay was also claimed by the Giants after splitting last season with Buffalo Bills and Carolina Panthers.
Clay had just one reception for 28 yards in four games before being released by the Bills. He was then claimed by the Panthers, recording five catches for 57 yards over nine games while returning a punt for a touchdown in a 35-27 win against the New York Jets on Nov. 26.
Defensive backs Antonio Hamilton, Kamrin Moore and Michael Jordan and center Spencer Pulley were also claimed by the Giants on Sunday.