The 31-year-old joins the Baggies on a two-year deal with a performance-related one year option after his contract with the Premier League champions expired.
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Irvine said: "I’m delighted we’ve been able to agree a deal with Joleon.
“He’s an England international and a proven Premier League player, who has been at the top level for a number of years.
“He’s got Premier League and cup medals to his name and has had a fantastic career so far.
“When we realised he was available and there was a realistic chance of getting him here we started to work very hard on it."
Lescott moved to Manchester City five years ago from Everton - where he worked with Irvine - in a £22 million deal and has represented England 26 times.
“Full credit to the board and everyone at the club for getting this done as quickly and effectively as they did," Irvine continued.
“I know Joleon and the type of personality and character he is, in addition to knowing what he can do as a player.
“He’s a great professional and he will add to a number of very good professionals who are already in the squad.
“It’s vitally important for me that we don’t go into a Premier League season with only three centre-backs at senior level – and I’m delighted with the four proven centre-backs we now have.
“Nobody is guaranteed a place in the team; I have to pick the best two, or sometimes we might play with a three, and I have to work out the ones who can do the best job in any particular game.”
Lescott will link up with his new team-mates at pre-season training on July 7 and is the Baggies' second signing of the transfer window, following in the footsteps of Craig Gardner, though Irvine remains keen to bolster his squad, with Manchester City's Jack Rodwell a potential target.
"I know Jack really well. He started training as a 16-year-old with the first team (at Everton)," Irvine told The Daily Mirror.
"Jack is a very good player. He has had some injuries but he was a Rolls-Royce as a kid.
"Manchester City bought him for a huge amount of money so I can't imagine he is someone who will be available for a small fee."