He has not played since September and his relationship with the club had broken down since the 2014-15 campaign.
"I`ve had a tough two years but everything happens for a reason. I`m mentally stronger now," Berahino said
"Now I am finally here I just can`t wait to start. For Stoke to show their faith in me is unbelievable," he added.
"On match fitness I am not there yet, but I am going to work hard to get myself back so I can help my new team-mates climb the table."
Stoke chief executive Tony Scholes said: "We`ve signed a young English striker who has already proven his ability in the Premier League.
"After a frustrating period he`s now desperately keen to reignite his career and we look forward to seeing him do that with us."
Berahino reacted angrily to a bid from Tottenham being turned down on transfer deadline day in summer 2015 and two months later tweeted that he would never play for West Brom again under then-chairman Jeremy Peace.
And in January 2015, he scored four goals but barely celebrated in what was interpreted as a sign of his growing disillusionment at the Hawthorns.
Speaking after Saturday`s 4-0 defeat by Spurs, West Brom boss - and former Stoke manager - Tony Pulis had said Berahino would not be sold "unless it is right for the club".
He added: "It has to be a two-way situation. That has always been the situation; we will not sell the lad because it suits him."
England Under-21 forward Berahino is the Potters` second signing of this transfer window after the loan deal for Derby keeper Lee Grant was made permanent.
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