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.
Sources have also told BBC Sport that Paul Trollope, Joleon Lescott and Conor Coady have been sounded out over potential roles in Edwards’ new-look backroom staff.
The potential additions of Lescott and Coady are rated as difficult to complete – particularly the latter who is still playing for Wrexham.
In a statement, Boro confirmed Adi Viveash will continue to take charge of first team while the club searches for a new manager.
It continued: “Rob had signed a three-year contract with ourselves during the summer, but after he made it clear he was keen to join Wolves, we reluctantly agreed a figure of compensation for both himself and one of his first team coaches, Harry Watling.
“On Saturday, the Riverside came together as one, players, staff, and supporters, and you could feel the sense of unity which is at the core of our club. To everyone involved, we would like to say thank you.”
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.”
