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.
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)
