One of the few incidents of note saw unlucky home striker Dave Kitson, who has still to score for Stoke, replaced by Richard Cresswell after falling awkwardly in the six-yard box and hurting his knee.
Despite the personnel change, the home side looked the more likely side to score against a West Brom side who have not won at Stoke since 1982.
Sidibe tested Carson with a superb 25-yard strike after some nifty footwork and Rory Delap showed there is more to his game than an outrageous long throw when he shot over the bar after an impressive run from midfield.
Miller had the visitors' only opportunity of the opening half but his clever dipping volley just cleared the bar.
Not surprisingly, the only other realistic chance of a breakthrough came via a Delap throw-in or home corner.
Mercifully, the second half provided slightly better fare and twice within five minutes of the restart the lively Tom Soares went close for the home side.
The former Crystal Palace midfielder hit a fizzing deflected strike that well saved by Carson, before heading tamely at the England keeper from 10 yards.
We have to stop run of defeats - Mowbray
Albion were also beginning to threaten more on the break, playing some crisp, flowing football when they got the chance.
But Stoke, who have won every league game in which they have scored first this season, broke the deadlock to take maximum points when Sidibe rose to guide a powerful header into the far corner after a wonderful piece of interplay saw Higginbotham get free on the left.
It was a goal that ensured a fourth successive home win for Stoke, a miserable 45th birthday for Albion boss Tony Mowbray and leaves the Baggies with just one point from their last 21.
Stoke manager Tony Pulis: "We had to work hard because West Brom are a decent team but nobody could argue that we deserved to win.
"Our goalkeeper Thomas Sorensen did not have a save to make.
"We got a good beating against Manchester United in our last game but the character the players have shown in this game was fantastic."
West Brom boss Tony Mowbray: "It was a poor game in which there were very few goalscoring chances.
"Stoke are hard to break down because their pitch is narrow and they get a lot of men behind the ball.
"Stoke is a very difficult place to come to and there are more illustrious teams than us that have lost here.
"We need to start turning some of these narrow defeats into points."
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