Solbakken topped a list of candidates to replace Mick McCarthy, who was sacked as manager in February, and will officially take charge on July 1.
The 44-year-old, a former Norway international, had been odds-on with most bookmakers to take charge at the relegated club.
Chief executive Jez Moxey told the club's official website: "Following talks with Stale Solbakken I am happy to confirm that he has agreed to become our new manager. He will be formally introduced to the media on Monday."
Interim manager Terry Connor had also been interviewed and it is understood Wolves want to retain his services in some capacity, even if it is not as manager.
Solbakken said: "I am delighted to be taking up the important challenge to manage Wolves. It's an exciting opportunity to manage a great club and I can't wait to get started."
The Norwegian has an impressive pedigree although his record took a knock last month when he was sacked by Cologne, who were then third from bottom of the Bundesliga.
But before that he had led Copenhagen to five Danish titles and into the Champions League for the first time.
Solbakken won 58 caps for Norway between 1994 and 2000 and briefly played in the Premier League with Wimbledon in the late 1990s.
He retired as a player after suffering a heart attack in training in 2001 while at FC Copenhagen, who were then managed by the new England boss Roy Hodgson.
Defender Roger Johnson told Sky Sports News: "I am not too sure. I don't know who the guy is to be honest. It is out of the blue. We'll see what happens."
The problems in Cologne were not just Stale Solbakken: he was just the easiest one to fire. The whole situation there was really hard this season. They needed to find a new chairman, they had a general manager (Volker Finke) who is a bit egocentric and does not allow other opinions than his and they had a team which was all focused on one person: Lukas Podolski. They all hid behind Podolski.
So in my opinion Solbakken is not the one to blame for Cologne's disappointment. He has a good sense of humour. For example, when Cologne lost one of his last games as coach, his mobile phone rang during the press conference. He looked at it and said: "It's my wife. She wants to ask if I still have a job tomorrow."
Tactically he tries to let the opponent attack only on the wings and likes to close up through the middle. The problem in Cologne was that his defence had problems in the centre when defending headers. Normally he does not play with a high pressing game but tries to keep his lines close together.