Irvine has had to wait patiently for Davidson to resolve a contract dispute with Dutch Eredivisie club Heracles Almelo.
Davidson, who played in Australia's three World Cup group matches, believed he had finished his contract with Heracles and had become a free agent.
Heracles, however, informed Davidson they had triggered a one-year option on that deal and he remained their player.
To get the deal over the line, Albion have paid an undisclosed fee for the 23-year-old left-back, who has signed a two-year contract, with an option in the club's favour for a further season.
Speaking to the club's website, Irvine said: "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.
"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."
Irvine added: "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."
Alongside Pocognoli, Davidson joins fellow new faces Craig Gardner, Joleon Lescott, Chris Baird, Brown Ideye and Andre Wisdom at The Hawthorns.
- Sports & Recreation