The 23-year-old striker has moved to the bet365.com Stadium on a five-and-a-half-year deal for a guaranteed fee of £12million, rising to £15million based on player and club success.
Berahino has made a total of 105 PL appearances for West Brom, scoring 23 goals.
"We've signed a young English striker who has already proven his ability in the Premier League,” Stoke chief executive Tony Scholes told stokecityfc.com of Berahino, their second signing of the transfer window after Lee Grant.
Berahino's move did not happen in time to be eligible for Stoke's match against Manchester United on Saturday.
That would mean he could make his debut on 1 February, at home to Everton.