The former Wolves defender, who has also had two separate spells at the club as a coach, penned a three-and-a-half year deal to succeed Vitor Pereira.
Wolves have agreed a compensation package with Middlesbrough to bring Edwards on board and the 42-year-old has released a statement addressing his decision to leave the Riverside Stadium.
'Wolves is a special club for me' - Edwards
Rob Edwards claps visiting fans from a previous visit to Molineux
Rob Edwards succeeded Vitor Pereira as Wolves' new head coach on Wednesday. (Nick Pots/PA)
"This has been one of the toughest decisions of my career," he wrote on Instagram.
"In truth, I wouldn't have considered leaving Middlesbrough for any other club.
"Wolves is a special club for me. It's where I grew up both as a player and a coach, it's still where my family calls home.
"It has always been my dream to come back and these opportunities do not come around too often."
Middlesbrough gratitude
Middlesbrough boss Rob Edwards has been linked with the Wolves hotseat after a fine start to life at Middlesbrough. (Photo by George Wood/Getty Images)
Edwards leave the Riverside Stadium with Middlesbrough sitting second in the Championship. (Photo by George Wood/Getty Images)
Edwards took over Middlesbrough back in the summer and collected wins in each of his first four games to win the Championship manager of the month award in August.
He leaves with Boro sitting second and five points adrift of the second-tier summit, although he was stood down from the dugout for their 2-1 win over Birmingham City last weekend.
"Middlesbrough is an incredible club, with a fantastic owner, staff and passionate supporters," he added.
"I want to thank everyone at the club for the opportunity to lead this squad.
"It is a great group of players that I genuinely hope and believe will deliver success this season.
"I could not be prouder of the environment we built together and how it all gelled together so quickly - the players, the staff and the fans, who drive it all.
"I sincerely wish everyone at Middlesbrough all the very best for the rest of the season and beyond."
Edwards takes over at Molineux with Wolves rooted to the foot of the Premier League table after picking up just two points from their first 11 games.
That dismal start has already left them eight points adrift of safety and without a league win since April 26.