It was one-way traffic in the first half, but Ollie Watkins' 14th minute finish was all the away side had to show for their dominance.
Clarets skipper Ben Mee drew the hosts level when converting Ashley Westwood's corner early in the second half.
Villa then restored their lead in the 68th minute when Jack Grealish got on the end of a Douglas Luiz cross.
Dwight McNeil picked out the far corner to equalise for a second time in the 76th minute.
And then Chris Wood netted his fourth goal in his last five starts against the Villans with a terrific header three minutes later.
Here are the RATINGS.
1. Nick Pope 9
Absolutely outstanding. Lost count of how many crucial stops the England international made. Watkins, Grealish, Traore, McGinn all denied. Deserves a huge amount of credit for keeping Burnley in the game.
2. Matt Lowton 6
Caught too narrow for Villa's opener when unable to get out to stop the cross from Targett and struggled to cope with the movement of Grealish at times. However, vastly improved after the break and Villa's productivity was stunted down that side.
3. Erik Pieters 6
His experience is invaluable. The Dutchman knows how to contend with different types and different styles, reading exactly when to commit and when to stand a player up. Villa's second came from his side, but produced another good display.
4. James Tarkowski 7
Not at his best in the first half as Villa got bodies beyond Burnley's middle bank. Caught out from Cash's ball over the top in the first half, but gave the home side's backline a shot in the arm after the break with a commanding performance.