Monken, 53, spent only one season on former Browns head coach Freddie Kitchens' staff, and new Cleveland coach Kevin Stefanski didn't retain him.
Monken will take over playcalling duties for the Bulldogs this coming season, after their offense struggled at times in 2019. Georgia was fifth in the SEC in scoring (30.8 points per game), total offense (408.1 yards), passing (223.0) and rushing (185.1).
Georgia assistant coach James Coley, who called offensive plays this past season, is expected to move into a new role. Former Ole Miss coach Matt Luke previously joined Georgia's staff to replace offensive line coach Sam Pittman, who left to become head coach at Arkansas.
Monken previously was the offensive coordinator of the Tampa Bay Buccaneers and Oklahoma State. He was the passing game coordinator at LSU and worked as an assistant at Louisiana Tech and Eastern Michigan.
He was the head coach at Southern Miss from 2013 to 2015, with his teams compiling a 13-25 record.
Georgia's offense will undergo a face-lift after quarterback Jake Fromm, tailback D'Andre Swift and three starting linemen left early for the NFL draft. Top receiver Lawrence Cager and tight ends Eli Wolf and Charlie Woerner are departing, and starting offensive lineman Cade Mays transferred to Tennessee.