The Potters arrived having lost their last three Barclays Premier League games and seven of their previous nine but had the better of a scrappy first half.
The match improved slightly after the break, but neither team could find a winner, with West Brom's Romelu Lukaku and Stoke's Peter Crouch unable to convert headers and Kenwyne Jones denied in injury-time.
Chris Brunt sent a speculative attempt into the stand before the Northern Irishman broke into the box and hit at Begovic from a promising position.
Ben Foster survived appeals for a handball outside the area when referee Mike Dean determined he had been fouled by Cameron Jerome.
Stoke midfielder Glenn Whelan, who had been on the receiving end of a high boot from James Morrison at the end of the first half, was substituted for Dean Whitehead at the interval.
Shortly after the restart, the in-form Lukaku - scorer of eight goals in his previous 11 Premier League games - got his head to a Thomas cross, but it bounced through to Begovic.
And within a minute, Crouch nodded the ball onto the roof of Foster's net from a Matthew Etherington delivery.
A series of substitutions failed to yield the breakthrough, but Ryan Shawcross did have sight of goal spoiled by the offside flag.
Jones was in a similar position in the box at the death - though this time onside - as he chested the ball down, but he could not put his shot beyond Foster.