Winston, 33, re-signed with the Bengals this offseason. He was fighting for a backup spot on the Bengals' offensive line.
Gilberry, 32, also had re-signed with the team this offseason. He has 34 sacks in nine seasons, including 2.5 for the Bengals last season.
The Bengals hope to put Elliott, who they drafted in the fifth round of the 2017 draft, on their practice squad. The decision to cut him means veteran Randy Bullock will be the Bengals' kicker this season.
Winston was re-elected in 2016 to a two-year term as the NFLPA's president.
He made headlines earlier this month when he told television station WCPO in Cincinnati that he doesn't care if the NFL "dies out" in 20 years because of a possible work stoppage in 2021. He suggested that players shouldn't care that a strike or lockout "might kill the goose that laid the golden egg."