The 27-year-old, who is out of contract at the Hawthorns, will formally complete his move on July 1 after signing a four-year deal, the club has confirmed.
Jones said: "I'm really happy to be here and I'm looking forward to getting back for pre-season and kicking on.
"I've played at the Stadium of Light a number of times, so I know what it's about and what great backing the club has.
"I've come to the training ground today and it's amazing. I knew I was signing for a massive club with everything in place to kick on up the league and carry on moving forward."
Shrewsbury-born Jones began his career at Crewe, and joined the Baggies, for whom he made more than 70 senior appearances, from Preston in July 2011.
The fact that Poyet's first close-season acquisition working alongside new director of football Lee Congerton is a British player with Barclays Premier League experience is significant with the Black Cats changing direction after a dismal summer 12 months ago.
Of the 14 signings made by then director of football Roberto De Fanti for manager Paolo Di Canio, 13 were from overseas, many of them with little or no background in English football, and a season of struggles duly followed.
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