The former Albion defender, 32, has signed a two-and-a-half year deal after Wolves' relegation rivals finally accepted their offer to buy him earlier this week.
Dawson will not be available to play today (Sunday) against Manchester City as he was not signed in time.
Wolves had a deal agreed to sign Dawson in the summer but West Ham pulled the plug after failing to bring in a replacement.
Sporting director Matt Hobbs said: “Everyone will be aware it’s a signing we tried to do last window and it fell through in the end because the replacement West Ham were looking at fell through. Craig’s wanted to come here for the last six months, so it’s something we’ve kept warm and kept in touch on.
“His family are in the north of England, so for him it’s great to get closer to them. He’s a player we’ve wanted for a while, so it was great to get it done, even if it’s six months later than we wanted, but it’s good to get here now.
“In my first interview, I talked about players who know the league are important. Personalities are important. We lost a few leaders in the summer with John (Ruddy), Conor (Coady) and Romain (Saiss).
“We’re bringing another leader into the group, which is always a positive, but if you look at our centre backs, they're all relatively young, not just in age, but also the number of games they’ve played in English league football, let alone the Premier League.
“The amount of games Craig has played, he can be a foil for those guys on and off the pitch. He’s going to be a very good player for our squad and will be someone our young centre halves will learn a lot from and will help develop those players.
“He was on our list from before, but he also played against Julen last season for West Ham against Sevilla. He’s a player Julen knew very well, so, like the players we’ve managed to do so far, it was a very quick discussion on Craig from both of our sides.”
Wolves were forced to cancel a summer loan move for Yerson Mosquera after Dawson's deal collapsed, but Wolves are now set to send the youngster out for some game time this month.
Hobbs added: “We want to get Yerson out on loan, to start getting some games. The manager has real belief in him and the tools he has. We brought him in, and he was unlucky with his two hamstring injuries, and then it wasn’t a great time to drop an inexperienced defender back into the team.
“He’s been a victim of circumstances, so for us it’s important to get him out on loan, so the fact we bring Craig in to be the player he’ll be, but also help the defenders in the building and allow Yerson to go out on loan is great for the football club.”