Gamboa, 24, played in all five matches as Costa Rica made a stunning run to the quarter-finals in Brazil before losing on penalties to the Netherlands.
The right-back has signed for an undisclosed fee from Norwegian side Rosenborg, penning a three-year deal after the Football Association approved his application for a work permit.
"Cristian is a player I was extremely impressed by during the World Cup," said West Brom manager Alan Irvine.
"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."
Davidson, who made three appearances for Australia at the World Cup, has signed a two-year contract with an option for a further year after resolving a contract dispute with former club Heracles Almelo.
Davidson, 23, thought that he was a free agent after coming to the end of his contract with Heracles, but the Dutch club informed him that they had taken up an option to extend his contract by a year.
As a result, West Brom were obliged to pay an undisclosed fee for the left-back.
"Due to contractual issues with his former club, there's been a delay in Jason coming," Irvine told the club website.
"This means he hasn't trained properly since the World Cup, so he's going to need a little bit of time to catch up before we can expect him to perform at the levels he's capable of.
"But he is a good defender who has lots of energy, gets forward well, and will give us competition for places at left-back with Sebastien Pocognoli."
West Brom have now signed nine new players during the close season, with Gamboa and Davidson following Pocognoli, Craig Gardner, Joleon Lescott, Chris Baird, Brown Ideye and Andre Wisdom to The Hawthorns.
Having avoided relegation by three points last season, West Brom open their 2014-15 campaign at home to Sunderland a week on Saturday.