Edwards leaves Middlesbrough second in the Championship table, five points behind leaders Coventry City.
Wolves, meanwhile, are rooted to the bottom of the top-flight table after a poor start to the season in which they have picked up just two points from their opening 11 league games.
Edwards will be joined at Wolves by Harry Watling, who takes on the role of assistant head coach and was part of his team at the Riverside Stadium.
Wolves executive chairman Jeff Shi said Edwards had shown “tactical awareness” when he was a youth coach at the club.
“After he took first-team jobs he started to grow his own identity, character and leadership,” Shi added.
“[Rob’s] a very good person, he knows the club very well, he knows the city, the fans and he is very talented.”