Ollie Watkins put Villa ahead in the 14th minute with a deft finish to turn in a low cross from Matt Targett and Dean Smith's side could have gone in it at the break with a more handsome lead had it not been for fine saves by Burnley keeper Nick Pope.
But Burnley, who ended Liverpool's 68-match unbeaten home run last week, fought back again as Dwight McNeil made it 2-2 with a low cross-shot from the left which went through a crowded area into the bottom corner.
Winger McNeil turned provider in the 79th minute, whipping in a cross from the left flank which Wood met with a perfectly angled header.
Burnley, 15th in the table, moved on to 22 points and Villa dropped to 10th on 29.
Prior to the match, Villa had gone on to win each of their nine Premier League games in which they had opened the scoring, while Burnley were the only top-flight side yet to recover a single point after conceding the first goal.
"We got the equaliser and had grit and determination to come back again. It was fantastic from the lads," said New Zealander Wood.
"It's all about character in our team and we've got an abundance of it. We like to show people what we are about and what Burnley are about."
Smith was left to rue his side's failure to convert those first-half opportunities.
"The game should've been over at half-time, the chances that we created, they were quality chances," he said.
"Pope has made some really good saves, but the game should've been out of sight. I've just said to the players, it's very tough to win any game in this league and we have started the season so well but you can't get three points the way we played tonight."