The Browns also named former Arizona Cardinals head coach Steve Wilks to be their defensive coordinator.
Monken interviewed for the head coach openings with the New York Jets, Green Bay Packers and Cincinnati Bengals this offseason. Wilks was let go by the Cardinals after compiling a 3-13 record in his only season in Arizona.
The 52-year-old Monken had been Tampa Bay's offensive coordinator for the past three years, after he served as Southern Miss' head coach from 2013 to '15.
Wilks, 49, was the defensive coordinator for the Carolina Panthers prior to joining the Cardinals.
In addition to adding Monken and Wilks, the Browns also hired Mike Priefer as special teams coordinator, James Campen as associate head coach/offensive line coach, Stump Mitchell as running backs/run game coordinator, Ryan Lindley as quarterback coach, Jody Wright as special assistant to the head coach, Jim Dray as offensive quality control coach and Tyler Tettleton as offensive quality control coach.
The Associated Press contributed to this report.