The former England U21 international agreed terms on the deal at the bet365 Stadium on Friday evening after the Potters came to an agreement with Albion.
The 23-year-old had been a target for Stoke for the past three transfer windows.
The fee could rise to £15m with add-ons but he will be unavailable for Saturday's clash with Manchester United.
Stoke boss Mark Hughes was delighted to finally land the 23-year-old, who hit 23 Premier League goals in 105 appearances for the Baggies.
Hughes told Stoke City Player: "It's been a bit of a saga but we have managed to get it over the line, thankfully.
"At times it appeared like it wasn't likely to happen but, as always with these kind of deals, you have to be patient and hope that all the work you have done throughout the talks pays off.
"It did appear like it was stalling somewhat and I suggested that this morning, but that was based on my knowledge of the deal at that time.
"Evidently, it gained a fair amount of speed this afternoon and we have been able to conclude the deal, which we are all absolutely delighted about."
Hughes believes Berahino can thrive with the Potters and added: "He has a lot of untapped potential clearly, he is a player of good talent and somebody who has a lot of growth in the game still to come.
"He has already displayed his talents in the Premier League and proven that he can score goals at this level, which was a real attraction not only to us, but also to a lot of other clubs who were interested in him.
"I just feel that he has the potential to be the striker that we have been looking for for a long time - he is capable of creating chances and he is certainly capable of converting chances too.
"We are going to give him the stage to perform now, and he needs to embrace that, which I am sure he will."
Hughes, meanwhile, insists playmaker Bojan Krkic has not expressed a desire to leave Stoke.
The Spaniard has only started six games all season and has been linked with a move to Middlesbrough.
Hughes said: "We saw something in the press saying that if there was an opportunity he'd like to leave. He hasn't said that to me.
"He's our player, he's involved in the squad tomorrow so he's part of what we're doing. I think there have been enquiries but nobody's come back with any bids - we wouldn't encourage them to be perfectly honest.
"If they do come in, maybe then Bojan or his advisor would want to discuss that. But we're not at that point at the moment."
Hughes welcomes his former club United with both sides in decent form.
The Potters have won their last two Premier League games while United's nine-match winning run was ended by a draw with Liverpool last time out.
Stoke will be confident of taking something having claimed seven points from nine against United at home in their last three meetings, while they also held Jose Mourinho's side to a draw at Old Trafford earlier this season.
"We're playing them when clearly they're on a good run of form and the team is shaping up to be what you envisage a Jose Mourinho team should look like," said Hughes.
"You can see signs that that's forming as we speak. It'll be a big test. It's maybe a game we can enjoy. There's an expectation that Man United will come here and make it difficult and maybe win.
"We'll clearly have something to say about that and have done in recent seasons. We can maybe express ourselves a little bit more and really have a go and that's what I'll be emphasising to my players."
It could be a big day for two strikers.
Wayne Rooney continues his pursuit of the one goal he needs to become United's all-time leading scorer while in-form Stoke frontman Peter Crouch netted his 99th Premier League goal in last weekend's victory over Sunderland.
Hughes said: "We hope it (Rooney's record-breaking goal) is not against us clearly, we'd much rather Crouchy got his 100th goal, which is really important to him. So hopefully that'll be the case and Wayne will have to wait one week longer."
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