Nelson, 28, signed with the Sabres as an undrafted free agent in 2016 and has since spent four seasons with the organization, skating in 93 games with Buffalo and 151 games with the AHL's Rochester Americans.
He spent the entirety of his 2019-20 season in Rochester, registering 12 points and a plus-18 rating in 48 games.
"Obviously, I would have loved to be in Buffalo," Nelson said in a release from the team. "Things didn't work out that way, which is fine. I was happy coming back to the (Americans) and trying to help out the best that I could."
Nelson is now on track to be an unrestricted free agent at the end of next season.