The 27-year old midfielder officially comes out of contract with the Black Cats on July 1 when he will link up with the Baggies on a free transfer.
Albion's football director Richard Garlick said: "Craig is someone we have followed for a long time and we are delighted to finally get him.
"We tried to bring him here in the January transfer window but it did not work out.
"We have continued to track him and he has expressed a real desire to be at this club following interest from elsewhere.
"He has the flexibility to play in a few positions across the midfield, scores goals, and can slot in at right-back if required.
"I think he will bring a freshness to our midfield area and his character will suit our dressing room.
"We know we have a big summer ahead of us and it is great to get our first signing done so early."
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