Baird has signed a one-year contract, which includes a further performance-related 12-month option, with the Premier League club.
The 32-year-old was available on a free transfer following the completion of a two-month contract at Burnley, where he helped the team secure a place in the top flight.
Baird, who has eight years of Premier League experience, has been capped 65 times for Northern Ireland and can play as a full-back, centre-back or defensive midfielder.
Speaking to Sky Sports News, he said: "Last year was a tough year for me but I'm glad it's over and I'm looking forward to new things with Albion."
West Brom head coach Alan Irvine said: “I’d like to officially welcome Chris to the club.
“Chris is a player with vast Premier League and international experience who reads the game well and has good positional sense.
“He can play in several positions and his versatility effectively makes your squad bigger. It’s important we have the right characters in the dressing room and Chris is certainly that.
“We have done our homework on him. I have spoken to a lot of different people who have either managed or played with Chris and everyone spoke very, very highly of him both as a player and a person.”
Baird will train with his new team-mates, including recent arrivals Joleon Lescott and Craig Gardner, for the first time on Monday.
Albion’s technical director Terry Burton said: “We see Chris as an ideal addition to the squad we are building for the new season.
“The great thing about Chris is that he can provide competition and cover for so many positions, particularly at full-back, where we’re still looking to strengthen both at left and right-back.”