Rowett was also handed a £2,000 fine for the incident that happened during the 1-0 home defeat to Birmingham on October 20.
The 44-year-old was sent to the stands by referee Roger East after Stoke had two penalty appeals turned down.
Stoke issued a statement on their website that read: "Gary Rowett will serve a one-match touchline ban with immediate effect after he admitted an FA misconduct charge and accepted the standard penalty.
"The Stoke City manager's language and/or behaviour in the 87th minute of the game against Birmingham City on Saturday [20 October 2018] was abusive and/or insulting and/or improper.
"He has also been fined £2,000.
"The club will not comment further on this subject."
The ban means that Rowett will be forced to take a seat in the stands when Stoke travel to Bristol City this weekend.