Inside linebacker Daryl Washington will serve a four-game suspension to open the season. He led the Cards in sacks last season with nine. Abraham will try to fill the void caused by the suspension.
Abraham, 35, led the Falcons last season with 10 sacks but had to cut back on his snaps in a bid to stay healthy. He wasn't much of a factor in the playoffs after injuring his left ankle in the final regular-season game.
He was released by Atlanta in March.
Abraham had 68.5 sacks with the Falcons after being acquired in a trade from the New York Jets in 2006. He has 122 sacks in his career.
The team also announced that rookie wide receiver Ryan Swope was retiring because of concussion issues. The Cardinals placed the former Texas A&M star on the reserve/retired list, meaning he would be Arizona's property if he returned to the sport. Swope, drafted by Arizona in the sixth round, set school records for receptions and yards receiving for the Aggies but was plagued by concussion issues and had a serious setback in the early summer.
Information from The Associated Press was used in this report.