The 31-year-old has completed the free transfer to The Hawthorns after successfully undergoing a Bank Holiday Monday medical.
Lindegaard, who has been given the No.21 shirt, said: “Since moving to England I’ve been impressed with how well West Brom have done in the Premier League.
“I had a good chat with Tony Pulis – I must call him the ‘gaffer’ now – and he convinced me it would be a good move for me and the club.
“From there things moved very quickly. I was very keen to get it done so I can just concentrate on the thing that matters most, the football.
“I am very happy to join and hope I can make a big contribution this season.”
Lindegaard becomes the third United player to join West Brom in a matter of months, following in the footsteps of Darren Fletcher and Jonny Evans.
The Denmark international will compete with Boaz Myhill for the right to start in goal for the Baggies with Ben Foster still recovering from a serious knee injury.
West Brom goalkeeping coach Jonathan Gould said: “We are bringing an experienced goalkeeper into the mix.
“Anders has an international pedigree, he’s a big guy and been part of a very good team at Manchester United.
“He knows Fletch and Jonny Evans already – I’m sure he will fit in very well.”
Lindegaard made 29 appearances for the Red Devils, having previously played for Danish club OB Odense and Norwegian outfit Aalesund.
Lindegaard was named 2010 Goalkeeper of the Year in both Denmark and Norway, with his performances for Aalesund convincing Sir Alex Ferguson to sign him for a reported fee of £3.5m.