West Brom complete double swoop
Both players arrive at The Hawthorns having impressed at the recent World Cup in Brazil.
Costa Rica international Gamboa, who can play on the right of defence and midfield, has signed a three-year deal after being granted a work permit by a Football Association appeal panel.
He joins for an undisclosed fee from Rosenborg, while Australia left-back Davidson has moved from Heracles and penned a two-year deal, with his new club holding an option for a third.
West Brom have now made eight signings since the end of last season and head coach Alan Irvine hailed the latest newcomers.
Discussing the capture of Gamboa, Irvine said: "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."
Left-back Davidson will compete with another recent signing, Sebastien Pocognoli, for a starting berth.
"I'm delighted we’ve been able to complete a deal for Jason. He had a fantastic World Cup when he performed very well against three top teams," added Irvine.
"Jason 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.
"Due to contractual issues with his former club, there's been a delay in Jason coming.
"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.
"I spoke to Tim Cahill about Jason and he recommended him very, very highly both as a player and character.
"Tim sees him as a young player who will do extremely well in England and develop even further."