Robinson sustained the injury during the team's preseason game against the Arizona Cardinals. No timetable has been announced for Robinson's return.
The 26-year-old Robinson signed a one-year, $1.1 million deal with the Chiefs this offseason after spending the first two years of his career with the Las Vegas Raiders.
Robinson was suspended for the first three games of last season following a DUI charge back in March after he pinned a woman between two cars while under the influence.
Robinson posted 1.5 sacks with 13 tackles in 16 games (three starts) in his two seasons with the Raiders. He recorded a sack on quarterback Clayton Tune during Saturday's matchup before fracturing his foot.
Kansas City also placed safety Deon Bush on season-ending injured reserve after tearing his Achilles tendon during the same game.