The result moves Burnley out of the bottom three up to 17th place, on five points from eight matches, while Palace drop to 10th on 13 after nine games.
Sean Dyche's Burnley, who went into the match without a home goal in this campaign, ended their barren run when Palace defender Cheikhou Kouyate made a mess of a high cross from Johann Berg Gudmundsson, the ball falling to Jay Rodriguez who fed Wood to drive home.
Palace had suffered a blow before the game with the news that their influential winger Wilfried Zaha had been ruled out after testing positive for COVID-19.
Roy Hodgson's visiting side enjoyed plenty of possession though, with Andros Townsend and Jordan Ayew both testing Burnley goalkeeper Nick Pope while Michy Batshuayi drilled the ball just wide after a run from deep.
But Burnley should have added a second goal after the break when Gudmundsson was put in clear on goal but the Iceland international blasted his shot against the bar.
Palace applied late pressure and Burnley had Pope to thank again in the final minute when he came out bravely to foil Christian Benteke from close range.
"Nick made a good save in the first half but that one at the end was brilliant. But he is a top keeper," said Dyche, who was relieved at finally claiming three points.
"We were showing signs. That's three clean sheets in the last five games and tonight we altered how we were operating in the attacking third," he added.
"We created two or three golden chances, we take one and it gets a bit nervy because we hadn't found that first win so to get that one done will do us the world of good".
Palace boss Roy Hodgson felt his team had the better of the contest, but paid the price for a big mistake.
"I thought we played well second half. An unfortunate first goal, a misplaced header, but apart from that we didn't allow them many chances to score a goal," said the former England manager.
"In the second half they stood firm and held out for their victory. An error like that is always a possibility, you have to accept it and then you've got the rest of the game to put it right but we started a bit too late."