Wood had been under the weather in the days leading up to the Magpies visit to Turf Moor, with a sickness bug.
But his 58th minute header from Ashley Westwood's pin-point far post corner got the Clarets back to rude health after three successive defeats.
Cork said, “It's a great goal to win it and I've told Woody he plays better when he's ill.
"We could have done with him being ill last week, but we've been brilliant at set-pieces all season.
"Westy's quality has always been there and we just have a set-up at the moment that works and we look a threat from any balls into the box."