Wood, who had struck the post in the second minute, put the Clarets in front in the 31st minute with a fine header from a Dwight McNeil corner.
Cardiff were furious when referee Mike Dean reversed a decision to award a penalty for handball by Ben Mee after consulting with his assistant.
Junior Hoilett then wasted a great chance for Cardiff, blasting high over the bar after being found unmarked inside the area.
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Burnley made sure of the win in stoppage time when Wood added his second goal, nodding home from close range after a winding run from winger McNeil.
(Reporting by Simon Evans; editing by Clare Fallon)
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