The Costa Rica defender, who had already agreed personal terms and passed a medical, joins from Norwegian club Rosenborg on a three-year contract.
A Football Association appeal panel granted a permit on Tuesday, and Gamboa is now confirmed as Albion's eighth close-season signing.
Head coach Alan Irvine said: "As I commented the other day, it would have been amazing if Cristian had not been granted a work permit and we’re obviously satisfied with the outcome.
"Cristian is a player I was extremely impressed by during the World Cup. We faced stiff competition from other Premier League clubs for his services but we've managed to convince Cristian that West Bromwich Albion is the best club for him to join.
"Cristian is a quick, aggressive defender with good energy levels. He gets up and down the pitch really well and we believe he will continue to improve."
Gamboa, 24, can play at right-back or on the right of midfield and started all five of Costa Rica's World Cup games this summer.
West Brom's technical director Terry Burton added: "Cristian had been mentioned to us before the World Cup. We'd had some reports from his time in Norway and obviously his performances in BrazIl then confirmed our views."