Villa had been 2-1 ahead with 14 minutes remaining before Dwight McNeil’s low cross evaded everyone for an equaliser and the midfielder then crossed for Wood to score Burnley’s third.
The visitors dominated the first half and had gone in front when Ollie Watkins flicked the ball into the net following Matt Targett’s low cross from the left.
After a number of fine saves by Clarets goalkeeper Nick Pope, the hosts levelled through Ben Mee’s header, only for Villa to regain the lead with Jack Grealish’s half-volley.
But Burnley, who had only scored five times in their previous eight home Premier League matches this season, found two more goals to ease their relegation concerns.
They remain 15th in the table, but are now nine points ahead of 18th-placed Fulham, who drew 0-0 at fellow strugglers Brighton.