Edwards, 24, played his first three NFL seasons with the Raiders after being a second-round pick in the 2015 draft. He recorded 71 tackles and 5.5 sacks over the course of his career.
The Florida State product started a career-high 14 games in 2017 but fell out of favor under new head coach Jon Gruden in Oakland.
Gruden gave Edwards first-team reps during the preseason, but the defensive end didn't impress much. He finished the preseason with five tackles and no sacks.
"He's in a competitive situation," Gruden told reporters of Edwards. "We're trying to find a spot for him. We brought in two inside pass-rushers (Hurst and Hall). Can he be the 3-technique? Can he be the 6-technique? Is he a base end, is he an open-side end?
"He's got a lot of versatility. We'll look at the tape. He's in a competitive situation, and we'll have to let you know here in the next day or so what we're going to do."
The Giants added six players in total on waivers. In addition to Edwards, they claimed wide receiver Kaelin Clay, center Spencer Pulley and defensive backs Antonio Hamilton, Mike Jordan and Kamrin Moore.
The Giants waived quarterback Davis Webb, guard John Jerry, defensive back William Gay, tight end Jerell Adams, wide receiver Hunter Sharp and defensive tackle Josh Banks to make room for the changes.