The 42-year-old has agreed a contract until 2029 with Wanderers after they sacked Vitor Pereira earlier this month.
It is reported Championship side Boro will receive around £3million ($A6.0 million) in compensation after Edwards only joined them in June on a three-year contract.
With Middlesbrough currently in second place, Edwards is swapping a promotion race for a relegation scrap.
Edwards, who played 111 times for Wolves between 2004 and 2008 and went on to be a youth coach and an interim boss there, will be hoping to transform their fortunes after nine defeats from their opening 11 matches has left them sitting eight points adrift of safety.
His first game in charge will come after the international break on November 22 when Wolves play at home to Crystal Palace.
Edwards will be joined by assistant head coach Harry Watling after the pair worked together at Middlesbrough, with the rest of his backroom staff to be confirmed in the near future.
Wolves chairman Jeff Shi said: "I know Rob very well and I have seen his growth in different jobs.
"He's a very good person, he knows the club very well, he knows the city, the fans and he is very talented.
"When he was a youth coach here, he showed his tactical awareness, but after he took first-team jobs he started to grow his own identity, character and leadership.
"We need to refresh the whole club with a new coach's philosophy, bringing his own identity and ideas, and we can build on that. We are at a new chapter for the club and Rob will be a key piece of that."
With AP
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