Anderson, who played in the first two games this season, tore a quad tendon in last week's victory over Cleveland.
The 6-foot-2, 275-pound Gilberry is a fifth-year player out of Alabama who played from 2008 through 2011 with Kansas City. He signed with Tampa Bay as an unrestricted free agent before this season but was released on Friday.
He saw action for Kansas City in 54 games, including one in the postseason, and played in every game from 2009-11. He has 14 career sacks, including a career-high seven in 2010 when the Chiefs won the AFC West.