Wolves are paying €14m to sign the Portugal U21 international.
Gomes has signed a contract until 2029 with a one-year option after completing his medical on Tuesday.
Wolves boss Gary O'Neil said: "I'm delighted, especially to get one in so early. I think in transfer windows it's very easy for everything to drag on and get done very late, but from a squad point of view - laying plans for next season - it's really important.
"The type of player Rodrigo is, he had a fantastic season last season, playing in many different positions and he's only 20 years old. For where we are as a club, he's the perfect type of signing for us. He's young and has potential."
O'Neil added: "He's not proven at Premier League level yet, but he's one we can definitely know we can get to that level - we can work with him and help him develop, proving himself in the toughest league in the world.
"I'm really looking forward to working with him, he's a great character - every time I've watched him, his enthusiasm and endeavour has drawn my eyes towards him, so I'm really pleased and he's a really good signing for us."
Sporting director Matt Hobbs said: "He's a player we've followed for most of the season, since he first went on loan.
"He's a player we know very, very well and have done an awful lot of work on but getting him in has been relatively smooth because Rodrigo was clear that he wanted to come here and is excited."