Peter Whittingham scored a header on 65 minutes after he met Craig Noone's cross at the back post to give his side the narrow win.
Both sides struggled to create anything of note during the first 45 minutes but Chris Brunt came close with a fre kick on 37 minutes, although his goal-bound effort was cleared by Stephen Caulker.
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The Bluebirds delivered plenty of crosses into the Baggies penalty area, and Frazier Campbell nearly got on the end of Peter Odemwingie's excellent delivery at the end of the half, but he could not quite reach the centre.
However, it was a cross that eventually made the breakthrough when Noone picked out his fellow winger Whittingham at the back post.