O'Shea, who is joining on a deal until at least 2027, had an eight million pounds release clause but West Brom have reportedly agreed to let him leave for seven million to generate much-needed income.
O'Shea leaves West Brom after playing 107 games for the club, including during their 2020/21 season in the top flight.
"It's great to welcome Dara to Burnley and we know he can play a big part in the progression of our football club," coach Vincent Kompany said in a statement.
"Strong at the back as well as confident moving forward, a leader with experience and knowledge of playing at the highest level."
The Dublin-born O’Shea came through the West Brom academy and had loan spells at Hereford United and Exeter City.
He made 37 appearances last season, scoring twice, after being hampered in the 2021-22 campaign by an ankle ligament injury that kept him sidelined for six months.
"I want to improve as a player first and foremost and I want to help the club achieve their goals," O'Shea said.
"I'm at the stage of my career now when I want to push myself. I want to be the best I can be, and this is the club that can help me do that and I can help them too."
Championship winners Burnley will take on Kompany's former club Manchester City in the season opener on August 11th.
