Branch was attempting to break up a pass in the end zone against Jimmy Graham when his right knee appeared to buckle and he immediately went down in pain.
Branch has put together an impressive 2017 season for Arizona and is in line for a new contract next year.
Branch, 30, is a former fourth-round pick of the Raiders back in 2008. He spent seven years in Oakland before joining the Chiefs for the 2016 season.
The Cardinals elected to sign Branch to a two-year, $10 million contract that included $5 million guaranteed last March. Arizona placed him on injured reserve before bringing him back during last season.
Branch agreed to a pay cut this past March and is in line to be an unrestricted free agent next year.
In 2017, Branch appeared in nine games and recorded 69 tackles, no interceptions, a forced fumble and five pass defenses.