Matthew Etherington's penalty put the visitors 1-0 up in the 55th minute after referee Chris Foy adjudged Scott Carson to have fouled Kenwyne Jones.
Substitute Jon Walters converted another spot-kick with five minutes of normal time remaining and then bagged his second at the death to make it three wins out of three for the Potters.
That takes Tony Pulis' side, only recently hovering around the relegation places, up to eighth in the table as they built on victories over Birmingham and Liverpool.
The result brings more disappointment for the Baggies, though, who have now lost four of their last five league fixtures.
Source: PA
Source: PA